Barcelona housing agency for studentsA year ago I went to Barcelona to study at the University of Technology of the city. Because I was in South America until late August, and would arrive in Barcelona in the week classes would start, I would have little time to look for an apartment. I tried my luck on the internet and found different agencies offering lodging and apartments for students.1

After some discussion with my parents, they told me they would pay my apartment, up until an amount of €350 per month. Besides they would give me an allowance of €50, that included possible other expenses, such as public transport.

This puts the first criteria for the selection process:

P ≤ 350 (including ‘gastos comunes’)

Because there was still quite a number of offers in this price range, I went talking with a friend of mine who studied in Barcelona the last semester. I wondered if he could give me some advice about where to live in Barcelona (B). He told me that by far the best place to live was in (the neighborhood of) ‘el Born’. Raval and ‘Gotico’ were also in the centre, and acceptable, but a slightly ‘dirty’ area. Another nice ‘barrio’, but a bit further from the centre, was Gracia. The university would best to be reached living in the Raval or Gotico neighborhood.

Because of the pretty big amount of possibilities, it was possible to let these criteria count. I would choose from the offered apartments in the selected areas, or somewhere close to it.

L = {Born v Raval v Gotico v Gracia}

There were still some other factors that would play a role in the process of choosing an apartment. First was the preference of having at least 2 roommates (R), and the more the better, until the number of 5. Secondly the size of the room (S) would count in the decision. Finally the extras it had to offer, such as balcony, private bathroom, cable television, etc.

Two conditions had to be met regarding these extra factors:

R ≥ 2S ≥ 12 m²

With al these criteria I was able to make a selection on what seemed best to me. Some five alternatives were finally selected. These alternatives were to be judged based on the following factors:

- Location (L)- Number of roommates (R)- Size room (S)- Extras (E)

All of these factors have to be explained in further detail. The factor price was not taken into account, because I was given a maximum, and below that maximum my parents would pay for it. We go on with the explanation of the factors that will count in the decision making process.

Location (L): The location, as mentioned earlier, has to be in the three mentioned neighborhoods. Still, I preferred living in ‘el Born’ above Gotico, Gotico above Raval, and living in Raval above living in Gracia. Even within a certain neighborhood, there are some aspects that influence the valuation of the location, such as the nearness of a park or metro station. That’s why I made a distinction between location, and accompanying external aspects. I used the following for this; every location has its own valuation based on its neighborhood, which can be increased in case of positive external aspects (negative aspects are never mentioned), with a maximum of 1 point increase. I assigned the following values for each neighborhood:

Gracia: 1,0Raval: 2,0Gotico: 2,5Born: 3,0

1 www.loquo.com, www.habitatgejove.com

Number of roommates (R): An important aspect for me is to live with people, so as a criterion I set the minimum of 2 persons, but I rather have more people living with me. The value does not increase linear with every extra roommate. I value an increase of one roommate higher when this is the first extra roommate above the two I already have. To be more specific, I value the first roommate as much as the two following. The second extra roommate I value a bit more than the third. In numbers you can see it as a value of 0,5 for the first extra roommate, 0,3 for the second, and 0,2 for the third. Four or more extra roommates do not add extra value for me. This is explained in the graph below.

# Roommates Value

00,10,20,30,40,50,60,70,80,9

1

2,0 3,0 4,0 5,0

Figure 1 - Value function # of roommates

Size room (S): Another important aspect, the size of the room, has the same characteristic. I want a room that counts at least 12 m2. I value the first extra 2 square meters as much as the following 4 (each 0,5). In the graph below the value function can be seen.

Size Value

00,10,20,30,40,5

0,60,70,80,9

1

12,0 13,0 14,0 15,0 16,0 17,0 18,0

Figure 2 - Value function Size

Extras (E): The factor extras is difficult to define, because it can exist of many things, such as large kitchen, washing machine, balcony, windows in room, cable television, Internet, etc. It merely represents the impression of the apartment besides the already mentioned factors as number of roommates and location. The value range of this factor is from 0 to 5, the highest meaning the most luxurious apartment one could think of, with all extras imaginable, and even more. The note 0 is given to an apartment that does not have anything extras, just provides the basic living, washing and kitchen facilities.

In this way all (for me) important aspects in the process of looking for a room in Barcelona are covered, without overlap or correlation between the criteria, hence they are preferentially independent.

Multi-objective Value AnalysisTo make the best choice, I used a Multi Objective Analysis approach. The following value function can be used to describe the decision-making process.

v(L,R,S,E) = wLvL(L)+ wRvR(R) + wSvS(S) + wEvE(E)

With this function the one can value different alternatives, depending on the level of importance or weight (w) of each criterion, and the value (v) given to the criterion. The selected alternatives have the following scores on the different criteria, as described in the table below. Note that these are the scores, that the possible external effects and extra number of roommates are already processed in it.

Alternative

Location # of roommates Size Extras

1 1,5 3 18 1,52 3,5 5 13,5 23 2,5 4 15 34 2,5 2 12 4

Table 1 - Alternatives

By using swing weights I determined the importance for each of the criteria, which have to sum up to 1. The smallest value increment is for the size criterion, then extra roommates, followed up by extras, and finally location.wL = 0,50wR = 0,15wS = 0,10wE = 0,25

This means that I find the location of my future apartment twice as important than the extras it offers, and even more important than the size and number of roommates. One should not forget that I already have made a selection, in which quite some desires have been satisfied. The weights are thus assigned to the different extra features of the apartment regarding location, size, number of roommates, and extras offered.

Problem SolutionWe have now come to the part where we are going to solve the problem. We use for this the spreadsheet program Microsoft Excel. First the different value functions for each of the criteria will be described, then the scores, and based on the weighted single dimensional values, a recommendation will follow. After this recommendation a robustness analysis will conclude this document.

Value functionsWithin the following tables;

- Mono answers the question whether preferences are monotonically increasing (higher amounts of X are better), or decreasing (lower values are preferred).

- ρ is the exponential constant, that can be calculated through determining the normalized mid-value and subsequently finding its normalized exponential constant R. Then the multiplication of R and the range of the particular criterion gives ρ.

o Only for the criterion R and S the value of ρ is finite, because for the other criteria the value function is linear. This means that for these criteria, I value every increase or decrease the same.

o For criterion S ρ is calculated as follows, with X the standardized mid-value. (X-Low)/(High-Low) 0,33; look up R in z0,5 table 0,677. ρ = 0,677 * (High-Low) = 4,062

- Are all criteria explained with an exponential single dimensional value function.- Show the range of the values (Low-High).- Base considers the particular weights belonging to each of the criteria.

Location # of roommates Size Extras

X Value X Value X Value X Value

Low: 1 2 0 Low: 12 Low: 1,5

High: 4 3 0,5 High: 18 High: 4

Mono: increasing 4 0,8 Mono: increasing Mono: increasing

Rho: infinite 5 1 Rho: 4,062 Rho: infinite

Base: 0,5 Base: 0,15 Base: 0,1 Base: 0,25

Table 2 - Value functions

Now the Weighted Single Dimensional Values are being calculates, which will conclude in the decision on which of the alternatives is the best choice. The values for size are calculated in the following manner:

VS(x) = (1 – e[-(High – x)/ρ])/(1 – e[-(High – Low)/ρ])

Multiplied with the weight (0,1) gives the weighted single dimensional values. In the diagram below all values are filled in, which gives a clear overview of the results.

Alternative Weighted Single Dimensional Values Scores

L R S E1 0,08 0,08 0,10 0,08 0,332 0,42 0,15 0,01 0,10 0,683 0,25 0,12 0,03 0,15 0,554 0,25 0,00 0,00 0,20 0,45

Table 3 - Weighted Single Dimensional Value functions

RecommendationClearly alternative number two is the one to prefer. Especially the highly favorable location gives this alternative the most value. In the graph below the different alternatives are compared with regards to the different scores per criterion. Besides location, the number of roommates is a very important criterion in favor of alternative 2.

0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80

1

2

3

4

L R S E

Figure 3 - Overview scores per criterion

Robustness AnalysisThe result of the analysis is based on personal desires regarding my future living situation. These preferences of course are not generic, they do not count for every person. Especially the high importance on location could be different for people. Therefore a sensitivity analysis is being done, to determine the impact on alternatives with different models. In the following graph the influence of changing the weight given on Location can be seen.

11

11

11

22

22

22

3 3 3 3 3 3

4 4 4 4 4 4

0,00

0,10

0,20

0,30

0,40

0,50

0,60

0,70

0,80

0,90

1,00

0 0,2 0,4 0,6 0,8 1

1

2

3

4

Figure 4 - Sensitivity Analysis

After this analysis a conclusion can be drawn regarding the sensitivity of the preferred option; alternative 2. Only when the importance of the location, expressed in its weight, is below 0,2, another alternative seems more attractive, number 3. Another conclusion is that, if the other weights don’t change significantly, alternatives 1 and 4 are not preferred. This is not the only possible way of doing a sensitivity analysis, there are many possibilities.

ConclusionsAccording to this analysis I choose apartment number 2, because of its excellent location, and its overall score. Other people might have other preferences, regarding the weights assigned to criteria, as well as the value range (and value assignment!). Therefore this analysis can be considered very personal, and maybe not useful for people with other ideas, other desires, etc. On the other hand, there is a possibility to personalize this analysis, using the same method, but choosing your own criteria, and assigning weights to these criteria. Because the analysis is done in Excel, and the scores, values and weights are all connected, one could do other analyses by just changing the parameters. This in itself could be a great service for housing agencies, letting people fill in their most important issues, preferences, and criteria, through a short questionnaire, and calculating the best five options for them.

One can say that the more criteria you put in an analysis, the better the outcome will be, because you don’t leave any wish, thought, preference or other possible issue out of it. But it can be very difficult to give the right weights to each and every criterion, and expect that it really represents the person or persons doing the analysis. I would therefore recommend, when doing an analysis as such, not taking into account more than the four most important criteria.

Thieme Hennis05 April, 2006Stud: 1052381