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Decision support tools based on multiattribute analysis involve the use of different types of variables. These variables are aimed at providing a framework that allows preferences to be quantified. This is particularly useful in the field of sustainability, where variables with different units are involved. One widely accepted framework for standardizing different units is the value function. Studies of value function are complex and frequently have limited physical meaning. In this context, this paper emphasizes the need to define a general equation that reflects the preferences of the decision maker in a clear and easily applied way. The paper proposes a new general equation that fulfils these requirements. By modifying certain parameters, this general equation represents the most commonly used relationships (linear, convex, concave and Sshaped). The proposed equation is finally applied to four variables used in the field of industrial buildings and sustainability.
This paper deals with the standardization of units used to express different variables (indicators, criteria or attributes) involved in a decision making process and its application in the context of industrial buildings and sustainability. This standardization is achieved using a scale of preference, or the degree to which a certain desired outcome is satisfied. Such a preference usually lacks an objective measure that directly reflects the priority or priorities of the decision makers and their numerical magnitude. In order to estimate such a magnitude, numerous proposals have been put forward [
Some proposals use fuzzy techniques, which analyze complex problems with incomplete and inaccurate information describing real situations [
Other approaches include using probability density functions [
With regard to the scales used to assign values, most discussion is centered on the use (or rejection) of negative values. The inclusion of negative values (using, for example, a scale from −1 to +1, as proposed in [
As far as the area of application is concerned, the value function is widely used in medicine [
Once the different methods for defining and modeling the value function have been set, the difficulty lies in its application, since the majority of cases do not provide a real, physical sense of the variables involved and therefore users who are not wellversed in mathematics may find them difficult to use.
The objective of this paper is to respond to this difficulty by presenting a simple and understandable method for constructing a general value function and applying it in the field of industrial buildings and sustainability. It should be a universal function that allows the decision maker's preferences to be stated directly by modifying a series of physically meaningful variables. Obviously, a specific problem will require the assessment of different variables (actually all the relevant ones) that, in the context of sustainability, should cover the project's whole life cycle. This assessment will vary depending on the viewpoint (building developer, owner, user,
Defining the value function requires measuring preference, or the degree of satisfaction produced by a certain alternative option for a measurement variable (indicator). Each measurement variable may be given in different units; therefore, it is necessary to standardize them into units of value or satisfaction, which is basically what the value function does. The method proposed rates satisfaction on a scale from 0 to 1, where 0 reflects minimum satisfaction (
To determine the satisfaction value for an indicator, the MIVES model [
Definition of the tendency (increase or decrease) of the value function.
Definition of the points corresponding to the minimum (
Definition of the shape of the value function (linear, concave, convex, Sshaped).
Definition of the mathematical expression of the value function.
A description of each of these stages is presented in the sections below.
The value function can be increasing or decreasing depending on the nature of the indicator (or measurement variable) to be evaluated. An increasing function is used when an increase in the measurement variable results in an increase in the decision maker's satisfaction. In contrast, a decreasing value function shows that an increase in the measurement unit causes a decrease in satisfaction (see
Examples of indicators with a decreasing tendency as applied to sustainable industrial buildings include economic cost, time of execution, or emissions to the environment. Examples of this type of indicator with an increasing tendency would be those that reflect the proportion of recycled materials used in the building, the degree of adaptability to the surroundings, the flexibility of different elements or components of the industrial building,
Other value functions will have a mixed tendency, that is, functions that increase at first but later decrease. This type of function is characteristic of indicators with two points of minimum satisfaction and one maximum between them, as explained in the following section.
The points of minimum and maximum satisfaction define the limits of the value function on the
These points are usually established according to three criteria: existing rules and regulations, experience with previous projects, and the value produced by the different alternatives with respect to the indicator. A description of each of these follows.
Rules and regulations. The measurement variables are regulated by existing standards and are therefore limited to the values given, or to the minimum and maximum values included within the interval defined by them. The limits that are defined according to this criterion are quite inflexible since they usually must be complied with. As an example, we can consider the minimum fire resistance time of a structure according to the type and dimensions of the structure. In this case the minimum satisfaction is located at the point of minimum fire resistance time established by the corresponding regulations and cannot be changed. In the majority of cases, only one limit (minimum or maximum) is defined (e.g., minimum strength, maximum content). Sometimes, however, both limits (minimum and maximum) may exist (for instance in the case of temperature).
Experience with previous projects. When information on measurement variables is not provided by rules and regulations, these values can be determined by experience, from historical data, from data found in the literature, or from data obtained from previous projects. The range of values is slightly more flexible than when complying with rules and regulations.
The value produced by the different alternatives with respect to an indicator. In this case, the limits of the value function are provided by the minimum and maximum values of the different alternatives with respect to an indicator. Consequently, if a new alternative appears, the limits of the function and the corresponding value of the indicators may change.
In the case of having alternatives that generate values for variables that fall outside the established limits, they can be disregarded if the minimum or maximum values cannot be surpassed (for example if they correspond to regulatory limits) or, alternatively, the value of the limit surpassed (0 or 1) can be assigned to them. Choosing one option over another clearly depends on the variable considered.
If there are two points of minimum satisfaction and only one maximum, as shown in
Given that so far two coordinate points, (
A concave curve is used when, starting from a minimum condition, satisfaction rapidly increases at first in relation to the indicator (see
A convex function is appropriate when there is hardly any increase in satisfaction for small changes around the point that generates minimum satisfaction (see
A linear function reflects a steady increase in the satisfaction produced by the alternatives (see
An Sshaped function is a combination of the concave and convex functions. A significant increase in satisfaction is detected at central values, while satisfaction changes little as the minimum and maximum points are approached (see
where
It can be seen that the shape of the function depends on the values that the parameters
When the specific shape of the value function for an indicator is unclear, it may be defined by a working group. When this is the case, several value functions (discrete or continuous) may initially be defined according to the proposals given by each or some of the members of the group for the measurement variable (indicator). This means that rather than a single function, a family of functions is obtained, as shown in
According to this figure, several values on the
In this section, the procedure for determining the value function of four of the indicators that may characterize the design of a sustainable industrial building, as proposed in [
Once a material has come to the end of its useful life in a building, it can be used again as a resource (reused or recycled) in other applications. This process reduces the impact on the environment (fewer natural resources required) and the use of energy (mining of natural resources saved). For this reason, this use is encouraged or enforced by the regulations of different countries [
In the following points, the stages defined in the previous sections for determining the value function are applied to this indicator.
As previously mentioned, the use of recycled material in concrete makes a positive contribution to the environment and improves sustainability. Therefore, the higher the percentage of recycled material used in a building, the higher the degree of satisfaction obtained. Consequently, the value function will have an increasing tendency.
For this indicator, the points of minimum and maximum satisfaction can be defined through the regulations in force where the building is going to be constructed. Since in some countries the use of recycled materials in concrete is not very common, the point of minimum satisfaction can be established as 0%. In relation to the maximum, it is frequent that the regulations define this value based on technical matters (strengths, durability, previous experience,
Given that the use of recycled materials in concrete is not very common in Spain, it is positive to encourage it and a concave value function is proposed, in which small initial improvements are highly valued. The increase in value of the function is therefore maximized near the point of minimum value (
Since a concave shape has been decided for the value function in the previous stage, a
This indicator is used to assess the total cost of the construction of the industrial building. It includes all the costs needed to complete the structure, including such aspects as materials, labor costs, machinery, facilities, transport,
Naturally, lower costs generally result in greater satisfaction. Therefore, a decreasing tendency function should be defined, so that as costs increase, satisfaction decreases.
The points of minimum and maximum cost satisfaction in terms of €/m^{2} can be defined by experience with previous projects and by average market values. Typical values found for the cost of building different types of structures may vary between 20 €/m^{2} and 200 €/m^{2}. Consequently, the minimum and maximum limits are set in this case at 20 and 200 €/m^{2} respectively. In more specific cases, smaller ranges can be defined. Cost is a typical indicator (generically with a high variability) in which the range can be adapted to the available alternatives. If this approach is adopted, the minimum and maximum costs would become the points of maximum and minimum satisfaction, respectively.
Given that the objective is to reduce the building costs, a convex value function is proposed. This type of function penalizes severely high and medium costs while significantly rewarding low costs. For this type of function the increase in value is maximized near the point of maximum satisfaction.
Since the value function is convex, a
This indicator reflects the temperature in the building related to the comfort of people working inside. Temperature control systems in the building should maintain body temperature at an appropriate level. To calculate the required temperature, we need to consider factors such as the type of work that is performed in the building, the clothes worn by the workers, and the season.
Since very low or very high temperatures are not acceptable (low satisfaction), this indicator will, in general, have an intermediate optimum. This leads to an indicator with a mixed tendency (increasing until the optimum, and then decreasing). In order to define one single tendency, a redefinition of the points of minimum and maximum satisfaction must be established. For example, if temperatures are very low (15 °C) or in contrast very high (30 °C), with regard to what is tolerable for human comfort, then two points of minimum satisfaction are generated while there is only one point of maximum satisfaction, which is reached at the average temperature: 23 °C (see
That said, it becomes necessary (as shown in the next section) to transform the extremes with respect to the optimum temperature. This does not necessarily need to be the average point between the extreme values given above. Thus, if the distance to the optimum is defined as the new variable, a decreasing tendency will apply, since as we move away from the optimum temperature, satisfaction decreases.
As explained above, there are three points for this type of function: two points of minimum satisfaction and one of maximum satisfaction (
The range of values resulting from this transformation is 0–8 °C, where 0 °C corresponds to the maximum satisfaction since it is the most comfortable temperature with respect to the average (22−23 °C), while 8 °C represents the least satisfactory temperature as it is the furthest from the optimum temperature (23 − 15 = 8 °C)/(30 − 23 = 7 °C).
A linear value function is proposed for this indicator because the distance from the optimum temperature may be considered proportional to the corresponding dissatisfaction.
In this case, the equation that defines the indicator is given by
This indicator quantifies the number of possible contractors capable of and interested in making an offer for the job according to the system of contracts used in a specific case [
For this indicator, a function with an increasing tendency is defined since as the number of contractors increases the range to choose from also increases with respect to certain evaluation criteria (cost, technical solutions, experience,
Following the recommendations given in [
However, when there are many suitable contractors (n > 6) there is little increase in satisfaction as it is already a large enough number to ensure that the client receives realistic proposals.
Based on these values, setting the points of minimum and maximum satisfaction at 0 and 10, respectively, is recommended. The first figure (0) would cover the case in which there are no suitable contractors. The second figure (10) takes into account that it is not usually realistic to have more than 10 candidates meeting the conditions of a contract.
An Sshaped function is proposed in this case. This function tends to level off as the minimum value is approached for responses of between 0 and 3 alternatives, and also as the maximum value is approached for more than 6 alternatives. For values between 3 and 6, which is the critical range, there is a sharp increase in satisfaction.
The equation that defines the mathematical expression for the value function of this indicator is defined by the following parameters
A process that allows step by step definition of value functions to standardize and quantify preferences for different quantitative measurement variables has been presented in this paper. A mathematical formulation based on a continuous and flexible equation that provides a straightforward and understandable way of modeling has also been proposed. The starting points for this methodology are the tendency of the function (increasing or decreasing), the minimum and maximum satisfaction coordinates,
The value function proposed is modeled through a mathematical expression that depends on three basic parameters. The variation of these parameters leads to value functions that are physically representative and easy to construct, even for nonexperts. This flexibility makes it a useful tool that can be applied to any situation requiring multicriteria analysis within a wide range of applications. The examples presented, relevant to the field of industrial buildings, confirm this flexibility in cases requiring different types of value functions (linear, concave, convex or Sshaped). Even cases presenting an intermediate optimum can be modeled with this methodology through the use of an alternative indicator measuring the distance from the optimum. However it does not cover some complex cases as, for instance, value functions with an intermediate, but asymmetrical, maximum, with more than a maximum or with more than an inflection point, which are, in any case, very unusual, in particular for industrial buildings and in the construction sector.
It is important to emphasize that the sustainability assessment of any project will need the compiling and selection of all the relevant variables covering its whole life cycle. The value function of each variable will objectivize its subjectivity in accordance with a specific viewpoint (building developer, owner, user,
Tendencies of the value function.
Function with two minimum points and only one maximum.
Different types of value functions.
Value function generated by a working group composed of different decision makers.
Value function for the indicator recycled material (
Value function for the indicator cost of building the structure (
Value function for thermal comfort (
Value function for the number of contractors (
Typical values of
Increasing function  

Function  
Linear  ≈ 0  ≈ 1  
Convex 

< 0.5  >1 
Concave 

>0.5  <1 
Sshaped 

0.2/0.8 >1  >1 
Decreasing function  
Function  
Linear  ≈A0  ≈1  
Convex 

<0.5  >1 
Concave 

>0.5  <1 
Sshaped 

0.2/0.8  >1 
The authors wish to acknowledge the aid received through a number of projects funded by the Spanish Government (Spanish Interministerial Science and Technology Commission; MAT 200204310, BIA200509163C0301, BIA200914171C0401, BIA201020789C0401).