Garage Location Selection for Public Transportation System in Istanbul

Problem Definition

In this study, the most appropriate garage location selection problem for IETT is investigated and a methodology is developed to solve it. The steps of this methodology are shown in Figure 1 and the used techniques are explained in the following subsections.

Figure 1 The proposed methodology.

Techniques

In the proposed methodology, fuzzy AHP is used to determine the criteria weights and crisp axiomatic design, fuzzy axiomatic design, and weighted fuzzy axiomatic design are used to select the most appropriate garage location.

Fuzzy AHP

Defining the complex and hard situations is very difficult with numerical expressions, so linguistic variables have to be used in such situations. The linguistic variables are words/sentences in languages.

To deal with vagueness of human thought, the fuzzy set theory is introduced by Zadeh which was oriented to the rationality of uncertainty due to imprecision or vagueness. A major contribution of fuzzy set theory is its capability of representing vague data. The theory also allows applying of mathematical operators and programming to the fuzzy domain. A fuzzy set is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership (characteristic) function, which assigns to each object a grade of membership ranging between zero and one.

In our study, Buckley’s Fuzzy AHP methodology is used to fuzzify the hierarchical analysis by allowing fuzzy numbers for the pairwise comparisons and find the weights of all criteria. In the following the fuzzy AHP steps are explained.

Step 1. Construct pairwise comparison matrices among all the elements/criteria in the dimensions of the hierarchy system. Assign linguistic terms to the pairwise comparisons by asking which is the more important of each two elements/criteria.

Step 2. Use geometric mean technique to define the fuzzy geometric mean and fuzzy weights of each criterion by Buckley as follows:

$\tilde{r}_{i}=\left(\tilde{a}_{i 1} \otimes \tilde{a}_{12} \otimes \cdots \otimes \tilde{a}_{i n}\right)^{1 / n}$

$\widetilde{w}_{i}=\widetilde{r}_{i} \otimes\left(\widetilde{r}_{1} \oplus \cdots \oplus \tilde{r}_{n}\right)^{-1}$

where $\tilde{a}_{i n}$ is fuzzy comparison value of criterion $i$ to criterion $n$; thus, $\tilde{r}_{i}$ is geometric mean of fuzzy comparison value of criterion $i$ to each criterion, and $\tilde{w}_{i}$ is the fuzzy weight of the $i$th criterion and can be indicated by a TFN, $\tilde{w}_{i} = (Lw_i, Mw_i , Uw_i)$. Here $Lw_i$, $Mw_i$, and $Uw_i$ stand for the lower, middle, and upper values of the fuzzy weight of the $i$ th criterion.

Step 3. Take $\widetilde{E}_{i j}^{k}$ to indicate the fuzzy performance value of evaluator $k$ towards alternative $i$ under criterion $j$, and all of the evaluation criteria will be indicated by $\widetilde{E}_{i j}^{k}=\left(L E_{i j}^{k}, M E_{i j}^{k}, U E_{i j}^{k}\right)$:

$\widetilde{E}_{i j}=\left(\frac{1}{m}\right) \otimes\left(\widetilde{E}_{i j}^{1} \oplus \widetilde{E}_{i j}^{2} \oplus \cdots \oplus \widetilde{E}_{i j}^{m}\right)$

The end-point values $LE_{ij}$, $ME_{ij}$, and $UE_{ij}$ can be solved by the method put forward by Buckley; that is,

$L E_{i j}=\frac{\left(\sum_{k=1}^{m} L E_{i j}^{k}\right)}{m} ; \quad M E_{i j}=\frac{\left(\sum_{k=1}^{m} M E_{i j}^{k}\right)}{m} ;$

$U E_{i j}=\frac{\left(\sum_{k=1}^{m} U E_{i j}^{k}\right)}{m}$

Step 4. The criteria weight vector $\widetilde{w}=\left(\widetilde{w}_{1}, \ldots, \widetilde{w}_{j}, \ldots, \widetilde{w}_{n}\right)^{t}$; the fuzzy performance matrix of each of the alternatives $\widetilde{E}=\left(\widetilde{E}_{i j}\right)$. The final fuzzy synthetic decision matrix
$\widetilde{R}=\widetilde{E} \circ \widetilde{w}$.

$\widetilde{R}_{i}=\left(L R_{i}, M R_{i}, U R_{i}\right)$, where $LR_i$, $MR_i$, and $UR_i$ are the lower, middle, and upper synthetic performance values of the alternative $i$ ; that is,

$L R_{i}=\sum_{j=1}^{n} L E_{i j} \times L w_{j} ; \quad M R_{i}=\sum_{j=1}^{n} M E_{i j} \times M w_{j}$

$U R_{i}=\sum_{j=1}^{n} U E_{i j} \times U w_{j}$

Step 5. The procedure of defuzzification is to locate the best nonfuzzy performance value (BNP). Utilize the COA (center of area) method to find out that the BNP is a simple and practical method, and there is no need to bring in the preferences of any evaluators, so it is used in this study. The BNP value of the fuzzy number $\tilde{R}_{i}$ can be found by the following equation:

$\mathrm{BNP}_{i}=\frac{\left[\left(U R_{i}-L R_{i}\right)+\left(M R_{i}-L R_{i}\right)\right]}{3}+L R_{i}, \quad \forall i$

Axiomatic Design (AD)

AD has been put forward as a "scientific design approach" since the early 1980s. However, it started to be used in various areas as a method of design engineering intensively after the issue of the book which has been written by E. Yılmaz, 2006.

In our study, crisp AD is used for crisp criteria, fuzzy AD is used for fuzzy criteria, and the weighted fuzzy AD is used to take into account the different weights of each criterion.

Crisp AD. AD is proposed to compose a scientific and systematic basis that provides structure to design process for engineers. The primarily goal of AD is to provide a thinking process to create a new design and/or to improve the existing design. To improve a design, the axiomatic approach uses two axioms named "independence axiom" and "information axiom".

Independence axiom states that the independence of functional requirements (FRs) must always be maintained, where FRs are defined as the minimum set of independent requirements that characterize the design goals. Then, information axiom states that the design having the smallest information content is the best design among those designs that satisfy the independence axiom.

The information axiom is a conventional method and facilitates the selection of proper alternative. In other words, information axiom helps the independence axiom to put forward the best design. The information axiom is symbolized by the information content that is related to the probability of satisfying the design goals. The information content (I) is given by

$I_i = log_2 \dfrac{1}{p_i}$

where $P_i$ is the probability of achieving a given FR.

In any design situation, the probability of success is given by what designer wishes to achieve in terms of tolerance (i.e., design range) and what the system is capable of delivering (i.e., system range). As shown in Figure 2, the overlap between the designer-specified "design range" and the system capability range "system range" is the region where the acceptable solution exists. Therefore, in the case of uniform probability, distribution function  $P_i$  may be written as

$P_i = \dfrac{\text{common range}}{\text{system range}}$

So, the information content is equal to

$I_i =  log_2 \dfrac{\text{system range}}{\text{common range}}$

The probability of achieving $FR_i$ in the design range may be expressed, if $FR_i$ is a continuous random variable, as

$p_{i}=\int_{d r_{i}}^{d r^{u}} p_{s}\left(\mathrm{FR}_{i}\right) d \mathrm{FR}_{i}$

where $p_s(FR_i)$ is the system pdf (probability density function) for $FR_i$. Equation (9) gives the probability of success by integrating the system pdf over the entire design. In Figure 3, the area of the common range ($A_{cr}$) is equal to the probability of success $p_i$. Therefore, the information content is equal to

$I_{i}=\log _{2}\left(\frac{1}{A_{\mathrm{cr}}}\right)$

Figure 2  Design range, system range, common range, and PDF of a FR.

Figure 3 The probability of success for a one-FR, one-DP design.

Fuzzy AD. MCDM techniques in the literature are solutions when data is not crisp. In addition, fuzzy multiple criteria AD approach can be used when data is not crisp.

The fuzzy data can be linguistic terms, fuzzy sets, or fuzzy numbers. If the fuzzy data are linguistic terms, they are transformed into fuzzy numbers first. Then all the fuzzy numbers (or fuzzy sets) are assigned crisp scores. The following numerical approximation systems are proposed to systematically convert linguistic terms to their corresponding fuzzy numbers. The system contains five conversion scales as in Figure 4.

Figure 4 Triangular fuzzy numbers.

In the fuzzy case, we have incomplete information about the system and design ranges. The system and design range for a certain criterion will be expressed by using "over a number," "around a number," or "between two numbers". Triangular or trapezoidal fuzzy numbers can represent these kinds of expressions. We now have a membership function of triangular or trapezoidal fuzzy number whereas we have a probability density function in the crisp case. So, the common area is the intersection area of triangular or trapezoidal fuzzy numbers. The common area between design range and system range is shown in Figure 5:

$I=\log _{2}\left(\frac{\text { TFN of System Design }}{\text { Common Area }}\right)$

Figure 5 Common area of system and design range.

Weighted Fuzzy AD. In the method in Section 4.2, the weights for all subcriteria are equal. If the decision maker wants to assign a different weight ($w_j$) for each criterion, the following weighted multiattribute AD approach can be used.

The following is proposed for the weighted multiattribute AD approach:

$\left[\log _{2}\left(\frac{1}{p_{i j}}\right)\right]^{1 / w_{j}}, \quad 0 \leq I_{i j} \leq 1$

$\left[\log _{2}\left(\frac{1}{p_{i j}}\right)\right]^{w_{j}}, \quad I_{i j} \geq 1$

$w_{j}, \quad I_{i j}=1$

where $I_{i j}$ is the information content of the alternative $i$ for the criterion $j$; $w_j$ the weight of the criterion $j$; $p_i j$ is the probability of achieving the functional requirement $FR_j$ (criterion $j$) for the alternative $i$.

The strength of the proposed method over the existing methods can be explained as follows. AD approach takes into account the design range of each criterion determined by the designer. Thus, the alternative providing the design ranges is selected in AD approach while the alternative meeting the criteria at their best levels is selected in many other methods.

For example, if the designer wants to satisfy the criterion called "technological infrastructure" in the determined design range by himself/herself, he/she can use AD approach. The designer may not want to meet this criterion at its best level because of its cost. This opportunity is not possible when most of the other existing methods such as AHP, fuzzy AHP, and scoring models are used. The AD approach also differs from many other existing methods from the point of the rejection of an alternative when it does not meet the design range of any criterion. However, the decision maker can assign a numerical value instead of "infinitive" in order to make the selection of an alternative which meets all other criteria successfully possible, except the criterion having an "infinitive" value.

Solving AD manually requires a lot of manual calculation. Thus an application called ADSolver was developed by authors. ADSolver was developed using AD algorithms on  .net framework 3.5 and MSSQL 2008 Server Database. ADSolver includes additional properties to solve both Crisp and Fuzzy Axiomatic Design problems. If the weights of criteria are different or equal, ADSolver can be used to solve the problem. The pseudocode of ADSolver is given in Pseudocode 1.

Pseudocode 1 Program ADSolver.

Algorithms of ADSolver