Extended prioritizing of store plan alternatives produced with shape grammar using the generalized Choquet integral method

Abstract

In the previous paper, the store plan alternatives produced with shape grammar were evaluated using two of multi-criteria decision making techniques with triangular fuzzy numbers, namely fuzzy Analytic Hierarchy Process and fuzzy ANP. In addition to fuzzy Analytic Hierarchy Process and fuzzy ANP results, another multi-criteria decision making technique, the generalized Choquet integral method with trapezoidal fuzzy numbers, is used for evaluation and the result of this method is compared with fuzzy Analytic Hierarchy Process and fuzzy ANP results. The main contribution of this paper is to determine the interdependency among main criteria and subcriteria, the nonlinear relationship among them and the environmental uncertainties while prioritizing plan alternatives using the generalized Choquet integral method with the experts’ same data that evaluated in the previous study. To the authors’ knowledge, this will be the first interdisciplinary study which uses the generalized Choquet integral method in the field of architectural design.

Keywords

Computer-aided design Choquet integral shape grammars store design decision making

Introduction and literature review

In the previous study (Ozdemir and Ozdemir, 2017), multi-criteria decision making techniques, fuzzy Analytic Hierarchy Process (AHP) and fuzzy ANP methods were used for the evaluation of store plan alternatives that produced with shape grammar. Then, the obtained results of these techniques were compared.

Analytic Hierarchy Process is one of the common methods with which to solve MCDM problems. The first application to solve MCDM problem in the literature was Saaty’s choosing a school for his son using AHP (Saaty, 1980). The AHP uses numerical evaluations of dual comparative comparisons to form a priority or weight for each criterion. In addition, the AHP allows decision makers to make qualitative decisions and makes it possible to make systematic decisions by expressing the interaction and hierarchy of factors, thus reducing a rough estimation risk (Saaty, 1980).

The decision problem is structured hierarchically at different levels in this methodology (Mikhailov, 2003). The local priorities in AHP are established using pairwise comparisons and judgments using a comparison matrix that is set up by comparing pairs of criteria or alternatives (Promentilla et al., 2008). To express decision makers’ preferences, values ranging from 1 (equally important) to 9 (extreme important) is used. For the final step, the priorities in the AHP are assessed indirectly from pairwise comparisons judgments (Mikhailov, 2003) and conduct synthesis of priorities to calculate a weight for each alternative based on comparisons (Kuo and Chen, 2006).

Schoner and Wedley (1989) showed that there is a necessary correspondence between the manner in which criteria importances are interpreted and computed and the manner in which the weights of the options under each criterion are normalized. In general, if this relationship is ignored, incorrect weights are generated for options under consideration regardless of whether new options are added or deleted.

Choo et al. (1999) identified several plausible interpretations of criteria weights and their appropriate roles in different multi-criteria decision making models by analyzing the aggregation rules, identifying partial values, specifying explicit measurement units and explicating direct statements of pairwise comparisons of preferences.

Zadeh (1965) introduced the fuzzy set theory to deal with the uncertainty due to imprecision and vagueness. A major contribution of fuzzy set theory is its capability of representing vague data. A triangular fuzzy number that defined as (l,m,u), where l ≤ m ≤ u, denote the smallest possible value, the most promising value and the largest possible value.

The fuzzy AHP technique can be viewed as an advanced analytical method developed from the traditional AHP. The AHP’s subjective judgment, selection and preference of decision-makers have great influence on the success of the method. The conventional AHP still cannot reflect the human thinking style. Avoiding these risks on performance, the fuzzy AHP, an extension of AHP with fuzzy numbers, was developed to solve the hierarchical fuzzy problems. Buckley extended Saaty’s AHP to the case where the evaluators are allowed to employ fuzzy ratios in place of exact ratios to handle the difficulty for people to assign exact ratios when comparing two criteria and derive the fuzzy weights of criteria by geometric mean method (Hsieh et al., 2004). Fuzzy AHP can be used for the evaluation and ranking of alternatives (Kahraman et al., 2004; Mikhailov and Tsvetinov, 2004; Rodríguez et al., 2013). In the F-AHP, to evaluate the decision-makers’ preferences, pairwise comparisons are structured using triangular fuzzy numbers.

In this paper, we apply the generalized Choquet integral method to select the best store plan alternative that produced with shape grammar. This is the first paper in the literature to apply this technique in the architectural design. The motivation for the research is the need of multi-criteria decision making method, which can handle the inner and/or outer dependencies and the interactions between the elements of a design network. For this reason, we aim at comparing the performance of Choquet integral with previous techniques AHP and ANP under fuzziness (Ozdemir and Ozdemir, 2017). We study the prioritizing of store plan alternatives produced with shape grammar.

The Choquet integral is a fuzzy integral with a numerical structure which is used to evaluate selection criteria by dividing them into parts. Successful establishment of a Choquet integral depends on results that the fuzzy criteria impose, which in turn establish the importance of each criterion or their combination (Yayla et al., 2013).

The Choquet integral has been used for the solution of multiple criteria decision-making problems in the literature. Chiou et al. (2005) proposed the non-additive fuzzy integral (Choquet integral) to cope with evaluation of fuzzy MCDM problems while there is dependence among considered criteria. The sustainable development strategy for aquatic product processors in Taiwan was investigated.

Meyer and Roubens (2006) presented a multiple criteria decision support approach in order to build rankings and suggest the best choice from a set of alternatives using the Choquet integral.

Tsai and Lu (2006) generalized the standard Choquet integral whose measurable evidence and fuzzy measures are real numbers. They showed that their proposed generalization can deal with fuzzy number types of measurable evidence and fuzzy measures.

Kong et al. (2007) presented a fuzzy Choquet integral approach to evaluate the capability of supplier and to deal with the supplier selection problem. Firstly, the basic of fuzzy measures and Choquet integral were introduced and then linguistic terms expressed in trapezoidal fuzzy numbers were used to assess the ratings for the suppliers’ quality and the importance of criteria. Lastly, an example was given to demonstrate their model.

Demirel et al. (2010) proposed a multi-criteria decision-making method using a fuzzy integral for the evaluation of alternative warehouse locations. They first determined the main and sub-criteria and the hierarchy for the warehouse location selection problem, then made a multi-criteria evaluation of the warehouse location alternatives to illustrate how the generalized Choquet integral can be used to do this.

Yayla et al. (2013) presented a case study on the selection of the optimal subcontractor for a Turkish textile firm using generalized Choquet integral methodology.

In this paper, the selection of the best store plan alternative in relation to the criteria determined in Ozdemir and Ozdemir (2017) from the alternative plans was handled as an MCDM problem and we used generalized Choquet integral methodology with the same data of Ozdemir and Ozdemir (2017) to solve the problem. Then the results of this methodology are compared with the results of the previously used methodologies, fuzzy AHP and fuzzy ANP (Ozdemir and Ozdemir, 2017). We aimed to find a useful way to handle fuzzy MCDM problems in a more flexible and more intelligent manner.

The rest of this paper is organized as follows: the problem definition is described in Section “Problem definition”. The generalized Choquet integral methodology is presented in Section “Choquet integral methodology”. In Section “Extended Application: Evaluation of Plan Alternatives”, we show an application of Choquet integral methodology in evaluation of plan alternatives. Computational results are given in this section. Finally, comparison of the results and future research directions are discussed in Section “Conclusion”, which concludes the paper.

Problem definition

Prioritizing the plans that produced with shape grammar (Ozdemir, 2014) was chosen for this study. In the previous study, authors asked three sector experts with the same importance weight (namely store owner, architect, and designer) about the problem of determining the best plan alternative. Three main criteria, eight sub-criteria and three alternatives were determined and weighted accordingly (Ozdemir and Ozdemir, 2017). In the numerical example, the store owner, the architect and the designer need to determine the best plan alternative for a store with three dead-walls that located in a shopping mall. For the problem, decision criteria and alternatives were defined by experts, as seen in Figure 1. In the previous paper (Ozdemir and Ozdemir, 2017), the main criteria were store presentation area features, store retail area features, and store–customer relationship. The arrows in Figure 1 represent the hierarchy of the problem.

Figure 1.

Hierarchy of the problem.

Store presentation area features criteria (C₁) include sub-criteria about presentation area: “Facade/Showcase (C₁₁)”, “Interior Layout (C₁₂)”, and “Entrance (C₁₃)”.

Store retail area features criteria (C₂) include sub-criteria about retail area: “Retail area position (C₂₁)”, “Security (C₂₂)” and “Circulation (C₂₃)”.

Store–customer relationship criteria (C₃) include the following sub-criteria: “Customer pleasure (C₃₁)”, “Items that meet the customer first (C₃₂)”.

As seen in Figure 1, the alternatives for plans were A1, A2, and A3 and these are given in Figures 2 to 4, respectively. In Figures 2 to 4, dead-walls are defined with “s” character and black cells, the character “c” represents the window, “t” is for the shopwindow-exhibit, “r” is for the interior-exhibit, “a” is for the cashier and the warehouse, and lastly “o” is for the changing room for customers (Ozdemir, 2014).

Figure 2.

Alternative A1 produced with shape grammar (Ozdemir, 2014).

Figure 3.

Alternative A2 produced with shape grammar (Ozdemir, 2014).

Figure 4.

Alternative A3 produced with shape grammar (Ozdemir, 2014).

In this study, the Choquet integral approach was used with the same data of Ozdemir and Ozdemir (2017). The results are compared with the previous results of fuzzy AHP and fuzzy ANP. The aim of using Choquet integral in this study and comparing with the previous results are to determine the interdependency among main criteria and subcriteria, the nonlinear relationship among them and the environmental uncertainties. Also using trapezoidal fuzzy numbers instead of triangular fuzzy numbers and the range computations using integral can enable better results for daily usage.

Choquet integral methodology

Choquet integral is a sort of general averaging operator that can represent the notions of importance of a criterion and interactions among criteria. A set of values of importance is composed of the values of a fuzzy measure. The success of a Choquet integral depends on an appropriate representation of fuzzy measures, which captures the importance of individual criterion or their combination (Demirel et al., 2010).

Relationship between trapezoidal fuzzy numbers and degrees of linguistic importance on a nine-linguistic-term scale can be seen from Table 1.

Table 1.

Relationship between trapezoidal fuzzy numbers and degrees of linguistic importance on a nine-linguistic-term scale (Delgado et al., 1998).

Low/high levels		Degrees of importance		Trapezoidal fuzzy numbers
Label	Linguistic terms	Label	Linguistic terms	Trapezoidal fuzzy numbers
EL	Extra low	EU	Extra unimportant	(0, 0, 0, 0)
VL	Very low	VU	Very unimportant	(0.00, 0.01, 0.02, 0.07)
L	Low	U	Unimportant	(0.04, 0.10, 0.18, 0.23)
SL	Slightly low	SU	Slightly unimportant	(0.17, 0.22, 0.36, 0.42)
M	Middle	M	Middle	(0.32, 0.41, 0.58, 0.65)
SH	Slightly high	SI	Slightly important	(0.58, 0.63, 0.80, 0.86)
H	High	HI	High important	(0.72, 0.78, 0.92, 0.97)
VH	Very high	VI	Very important	(0.93, 0.98, 0.98, 1.00)
EH	Extra high	EI	Extra important	(1, 1, 1, 1)

The methodology is composed of eight steps (Chen and Tzeng, 2001; Chiou et al., 2005; Demirel et al., 2010; Meyer and Roubens, 2006; Tsai and Lu, 2006):

Step 1: Given criterion i, respondents’ linguistic preferences for the degree of importance, perceived performance levels of alternative plans, and tolerance zone are surveyed.

Step 2: In view of the compatibility between perceived performance levels and the tolerance zone, trapezoidal fuzzy numbers are used to quantify all linguistic terms. Given respondent t and criteria i, linguistic terms for the degree of importance are parameterized by ${\tilde{A}}_{i}^{t} = (a_{i 1}^{t}, a_{i 2}^{t}, a_{i 3}^{t}, a_{i 4}^{t})$ , perceived performance levels by ${\tilde{p}}_{i}^{t} = (p_{i 1}^{t}, p_{i 2}^{t}, p_{i 3}^{t}, p_{i 4}^{t})$ , and the tolerance zone by ${\tilde{e}}_{i}^{t} = (e_{i 1 L}^{t}, e_{i 2 L}^{t}, e_{i 3 U}^{t}, e_{i 4 U}^{t})$ .

Step 3: Average ${\tilde{A}}_{i}^{t}$ , ${\tilde{p}}_{i}^{t}$ and ${\tilde{e}}_{i}^{t}$ into ${\tilde{A}}_{i}$ and ${\tilde{e}}_{i}$ respectively using equation (1).

{\tilde{A}}_{i} = \frac{\sum_{t = 1}^{k} {\tilde{A}}_{i}^{t}}{k} = (\frac{\sum_{t = 1}^{k} a_{i 1}^{t}}{k}, \frac{\sum_{t = 1}^{k} a_{i 2}^{t}}{k}, \frac{\sum_{t = 1}^{k} a_{i 3}^{t}}{k}, \frac{\sum_{t = 1}^{k} a_{i 4}^{t}}{k})

(1)

Step 4: Normalize the plan value of each criterion using equation (2).

{\tilde{f}}_{i} = \underset{α \in [0, 1]}{‖ {\bar{f}}_{i}^{α}} = \underset{α \in [0, 1]}{‖ [f_{i α}^{-}, f_{i α}^{+}]}

(2)

where

f_{i} \in F (S)

is a fuzzy-valued function.

\tilde{F} (S)

is the set of all fuzzy-valued functions

f, f_{i}^{α} = [f_{i α}^{-}, f_{i α}^{+}] = \frac{{\bar{p}}_{i}^{α} - {\bar{e}}_{i}^{α} + [1, 1]}{2}, {\bar{p}}_{i}^{α}

and

{\bar{e}}_{i}^{α}

are α-level cuts of

{\tilde{p}}_{i}

and

{\tilde{e}}_{i}

for all α = [0,1].

Step 5: Find the plan value of dimension j using equation (3).

(C) \int \tilde{f} d \tilde{g} = \underset{α = [0, 1]}{‖ ⌊ (C) \int f_{α}^{-} d g_{α}^{-}, (C) \int f_{α}^{+} d g_{α}^{+} ⌋}

(3)

where

{\bar{g}}_{i} : P (S) \to I (R^{+})

{\bar{g}}_{i} = [g_{i}^{-}, g_{i}^{+}]

{\bar{g}}_{i}^{α} = [g_{i, α}^{-}, g_{i, α}^{+}]

{\bar{f}}_{i} : S \to I (R^{+})

, and

f_{i} = [f_{i}^{-}, f_{i}^{+}]

for i = 1, 2, 3, …, n_j.

To be able to calculate this location value, a λ value and the fuzzy measures g(A_(i)), i = 1, 2, 3, …, n are needed. These are obtained from the following equations (4) to (6).

g (A_{(n)}) = g ({s_{(n)}}) = g_{n}

(4)

g (A_{(i)}) = g_{i} + g (A_{(i + 1)}) + λ g_{i} g (A_{(i + 1)}), where 1 \leq i < n

(5)

\begin{matrix} 1 = g (S) = {\begin{matrix} 1 / λ {Π_{i = 1}^{n} [1 + λ g (A_{i})] - 1} & if λ \neq 0 \\ \sum_{i = 1}^{n} g (A_{i}) & if λ = 0 \end{matrix} \end{matrix}

(6)

where A_i ∩ A_j = Ø for all i, j = 1, 2, 3, …, n and i ≠ j, and λ∈(−1,∞].

Let µ be a fuzzy measure on (I,P(I)) and an application $f : I \to R^{+} .$ The Choquet integral of f with respect to µ is defined by:

(C) \int_{I} fd μ = \sum_{i = 1}^{n} (f (σ (i)) - f (σ (i - 1))) μ (A_{(i)})

(7)

where σ is a permutation of the indices in order to have

\begin{matrix} f (σ (1)) \leq \dots \leq f (σ (n)), A_{(i)} = {σ (i), \dots, σ (n)} and \\ xf (σ (0)) = 0, byconvention . \end{matrix}

Under rather general assumptions over the set of alternatives X, and over the weak orders ${\underline{≻}}_{i},$ there exists a unique fuzzy measure µ over I such that (Demirel et al., 2010):

\forall x, y \in X, x \underline{≻} y \Leftrightarrow u (x) \geq u (y)

(8)

where

u (x) = \sum_{i = 1}^{n} [u_{(i)} (x_{(i)}) - u_{(i - 1)} (x_{(i - 1)})] μ (A_{(i)})

(9)

which is simply the aggregation of the monodimensional utility functions using the Choquet integral with respect to µ.

Step 6: Aggregate all dimensional performance levels of the plan alternatives into overall performance levels, using a hierarchical process applying the two-stage aggregation process of the generalized Choquet integral (equation (10)). The overall performance levels yield a fuzzy number, $\tilde{V} .$

\begin{matrix} main {criterion}_{(1)} = (C) \int fdg \\ \dots \\ main {criterion}_{(m)} = (C) \int fdg \end{matrix} á V = (C) \int main criterion \begin{matrix} dg \end{matrix}

(10)

Step 7: Assume that the membership of $\tilde{V}$ is µ_v(x); defuzzy the fuzzy number $\tilde{V}$ into a crisp value v using equation (11) and make a comparison of the overall performance levels of alternative plans.

F (\tilde{A}) = \frac{v_{1} + v_{2} + v_{3} + v_{4}}{4}

(11)

Step 8: Compare weak and advantageous criteria among the plan alternatives using equation (1) (Demirel et al., 2010).

Extended application: Evaluation of plan alternatives

In this paper, we apply Choquet integral methodology to select the best store design alternative produced with shape grammar. Then, the obtained result of this technique is compared with the previous results (Ozdemir and Ozdemir, 2017). The layout of the application case can be seen from Figure 5.

Figure 5.

The flow diagram of the application of Choquet integral method.

To solve the problem using Choquet integral, we make the comparisons with experts using trapezoidal fuzzy numbers as shown in Table 1 and the average values can be seen in Table 2. A1, A2, and A3 indicate the Alternative 1, Alternative 2, and Alternative 3, respectively. Average values on this table are the same with Table 2 of the previous study (Ozdemir and Ozdemir, 2017).

Table 2.

Average values used in Choquet integral.

Criteria	Importance value	Tolerance zone		A1	Tolerance		A2	Tolerance		A3	Tolerance
Criteria	Importance value	Importance value		A1	Zone A1		A2	Zone A2		A3	Zone A3
C1	EH
c11	VH	VH	EH	VH	VH	EH	M	M	H	M	M	H
c12	VH	VH	EH	VH	VH	VH	VH	VH	VH	VL	EL	SH
c13	VH	VH	EH	VH	H	EH	VL	EL	L	VL	EL	M
C2	EH
c21	VH	VH	EH	VH	VH	VH	H	H	VH	SH	M	VH
c22	H	H	EH	VH	H	EH	SH	SH	VH	M	SL	VH
c23	VH	VH	EH	VH	VH	EH	SH	SH	VH	L	L	SH
C3	VH
c31	VH	VH	EH	VH	VH	VH	SH	M	VH	EL	EL	M
c32	M	SL	EH	SH	L	EH	SH	SH	VH	SH	SL	VH

Average trapezoidal fuzzy numbers are used to quantify the linguistic terms in Table 2 (equation (1)). The tolerance zones in this table are obtained in that way: the first two numerical values of the lower linguistic value of a tolerance zone in Table 2 are combined with the last two numerical values of the upper linguistic value of the same tolerance zone. Consider the tolerance zone [M, H]. The corresponding numerical values of M and H are (0.32, 0.41, 0.58, 0.65) and (0.72, 0.78, 0.92, 0.97), respectively. Then the combined tolerance zone is (0.32, 0.41, 0.92, 0.97).

Table 3 gives the evaluation results by the generalized Choquet integral for α = 0. “Individual importance” column are the lowest and the highest value of the “importance value”. For example, the trapezoidal fuzzy numbers for “Very High” is (0.93, 0.98, 0.98, 1.00), and the “individual importance” for that criterion is obtained as (0.93, 1.00). For the subcriteria, equation (2) is used while equation (3) is for the main criteria. For example, the value [0.465, 0.535] of “A1 and subcriterion c11” is obtained in that way:

\begin{matrix} f, f_{i}^{α} = [f_{i α}^{-}, f_{i α}^{+}] = \frac{[0.93, 1] - [1 - 0.93] + [1, 1]}{2} = [0.465, 0.535] \end{matrix}

Table 3.

Evaluation results using the generalized Choquet integral for α = 0.

α = 0	Individual importance		Normalized plan alternative performance
α = 0	Individual importance		A1		A2		A3
C1	1.000	1.000	fi⁻	fi⁺	fi⁻	fi⁺	fi⁻	fi⁺
c11	0.930	1.000	0.465	0.535	0.160	0.360	0.160	0.360
c12	0.930	1.000	0.465	0.535	0.465	0.535	0.000	0.070
c13	0.930	1.000	0.465	0.535	0.000	0.070	0.000	0.070
C2	1.000	1.000
c21	0.930	1.000	0.465	0.535	0.360	0.520	0.290	0.465
c22	0.720	0.970	0.465	0.640	0.290	0.570	0.160	0.465
c23	0.930	1.000	0.465	0.535	0.290	0.465	0.020	0.150
C3	0.930	1.000
c31	0.930	1.000	0.465	0.535	0.290	0.465	0.000	0.035
c32	0.320	0.650	0.290	0.845	0.290	0.845	0.290	0.845

Table 4 gives the evaluation results by the generalized Choquet integral for α = 1. For the sub-criteria, equation (2) is used again. For example, the value [0.490, 0.500] of “A1 and subcriterion c11” is obtained in that way:

\begin{matrix} f, f_{i}^{α} = [f_{i α}^{-}, f_{i α}^{+}] = \frac{[0.98, 0.98] - [1 - 0.98] + [1, 1]}{2} = [0.490, 0.500] \end{matrix}

Table 4.

Evaluation results using the generalized Choquet integral for α = 1.

α = 1	Individual importance		Normalized plan alternative performance
α = 1	Individual importance		A1		A2		A3
C1	1.000	1.000	fi⁻	fi⁺	fi⁻	fi⁺	fi⁻	fi⁺
c11	0.980	0.980	0.490	0.500	0.205	0.300	0.205	0.300
c12	0.980	0.980	0.490	0.500	0.490	0.500	0.005	0.020
c13	0.980	0.980	0.490	0.500	0.005	0.020	0.005	0.020
C2	1.000	1.000
c21	0.980	0.980	0.490	0.500	0.390	0.470	0.315	0.410
c22	0.780	0.920	0.490	0.600	0.315	0.510	0.205	0.400
c23	0.980	0.980	0.490	0.500	0.315	0.410	0.050	0.100
C3	0.980	0.980
c31	0.980	0.980	0.490	0.500	0.315	0.410	0.000	0.010
c32	0.410	0.580	0.315	0.790	0.315	0.790	0.315	0.790

By solving the following equation for λ (Table 5), the fuzzy measures

g (A_{(i)})

, i = 1, 2, …, n are obtained using (equation (4) to (6)) as follows:

λ = \frac{{1 + [0.93 x (- 0.99965)]} x {1 + [0.93 x (- 0.99965)]} x {1 + [0.93 x (- 0.99965)]} - 1}{(- 0.99965)}

Table 5.

Fuzzy measures and λ values for α = 0.

			A1		A2		A3
	λ = 1.000002	λ = 1.000004	gi⁻	gi⁺	gi⁻	gi⁺	gi⁻	gi⁺
C1	−0.99965	−0.999996	0.463	0.535	0.452	0.535	0.319	0.650
c11			0.995	1.000	0.995	1.000	0.930	1.000
c12			1.000	1.000	0.930	1.000	0.995	1.000
c13			0.930	1.000	0.995	1.000	0.995	1.000
	1.00001	1.00001
C2	−0.99856	−0.99999	0.463	0.745	0.404	0.568	0.413	0.465
c21			0.996	1.000	0.981	1.000	0.930	1.000
c22			1.000	1.000	0.720	0.970	0.981	1.000
c23			0.930	1.000	0.981	1.000	0.996	1.000
	1.00002	1.000007
C3	−0.8399999	−0.9999899	0.628	1.155	0.290	1.092	0.093	0.562
c31			0.930	1.000	1.000	1.000	1.000	1.000
c32			1.000	1.000	0.320	0.650	0.320	0.650

That is,

\begin{matrix} λ = 1.000002 \\ g (A_{(3)}) = g_{3} = 0.930 \\ g (A_{(2)}) = g_{2} + g (A_{(3)}) + {λ g}_{2} g (A_{(3)}) = 1.000 \\ g (A_{(1)}) = g_{1} + g (A_{(2)}) + {λ g}_{1} g (A_{(2)}) = 0.995 \end{matrix}

Tables 5 and 6 summarize the whole fuzzy measures and λ values (for α = 0 and α = 1), which are computed in the example above.

Table 6.

Fuzzy measures and λ values for α = 1.

			A1		A2		A3
	λ = 1.000002	λ = 1.00009	gi⁻	gi⁺	gi⁻	gi⁺	gi⁻	gi⁺
C1	−0.99999	−0.9999	0.490	0.500	0.486	0.494	0.005	0.579
c11			1.000	1.000	1.000	1.000	0.980	0.980
c12			1.000	1.000	0.980	0.980	1.000	1.000
c13			0.980	0.980	1.000	1.000	1.000	1.000
	1.00001	1.00007
C2	−0.9999	−0.9999	0.490	0.698	0.447	0.505	0.423	0.418
c21			1.000	1.000	0.996	0.998	0.980	0.980
c22			1.000	1.000	0.780	0.920	0.996	0.998
c23			0.980	0.980	0.996	0.998	1.000	1.000
	1.00005	1.00001
C3	−0.9705	−0.9852	0.662	1.064	0.315	1.010	0.129	0.462
c31			0.980	0.980	1.000	1.000	1.000	1.000
c32			1.000	1.000	0.410	0.580	0.410	0.580

The aggregated Choquet integral values for the main criterion “C1” are calculated as in the following (equation (7) to (10)): In Table 6, the fuzzy measures for “A1” according to the main criteria are (0.490, 0.490, 0.662). The first overall plan alternative value for “A1” (0.318) in Table 7 is obtained in that way:

\begin{matrix} [1 x 0.662] + [1 x (0.49 - 0.662)] + [1 x (0.49 - 0.662)] = 0.318 \end{matrix}

Table 7.

Defuzzified overall values of alternative plans using generalized Choquet integral.

Criteria		Plan alternative performance			Defuzzified
		A1	A2	A3	A1	A2	A3
Overall plan alternative value		(0.318;0.616; 1.775;1.827)	(0.481;0.519;1.059;1.537)	(0.187;0.462;0.547;0.562)	1.134	0.899	0.440
C1	(0.463;0.49;0.5;0.535)		(0.452;0.486;0.494;0.535)	(0.005;0.319;0.579;0.65)	0.497	0.492	0.388
c11	(0.465;0.49;0.5;0.535)		(0.16;0.205;0.3;0.36)	(0.16;0.205;0.3;0.36)	0.498	0.256	0.256
c12	(0.465;0.49;0.5;0.535)		(0.465;0.49;0.5;0.535)	(0;0.005;0.02;0.07)	0.498	0.498	0.024
c13	(0.465;0.49;0.5;0.535)		(0;0.005;0.02;0.07)	(0;0.005;0.02;0.07)	0.498	0.024	0.024
C2	(0.463;0.49;0.698;0.745)		(0.404;0.447;0.505;0.568)	(0.413;0.418;0.423;0.465)	0.599	0.481	0.430
c21	(0.465;0.49;0.5;0.535)		(0.36;0.39;0.47;0.52)	(0.29;0.315;0.41;0.465)	0.498	0.435	0.370
c22	(0.465;0.49;0.6;0.64)		(0.29;0.315;0.51;0.57)	(0.16;0.205;0.4;0.465)	0.549	0.421	0.308
c23	(0.465;0.49;0.5;0.535)		(0.29;0.315;0.41;0.465)	(0.02;0.05;0.1;0.15)	0.498	0.370	0.080
C3	(0.628;0.662;1.064;1.155)		(0.29;0.315;1.01;1.092)	(0.093;0.129;0.462;0.562)	0.877	0.677	0.311
c31	(0.465;0.49;0.5;0.535)		(0.29;0.315;0.41;0.465)	(0;0;0.01;0.035)	0.498	0.370	0.011
c32	(0.29;0.315;0.79;0.845)		(0.29;0.315;0.79;0.845)	(0.29;0.315;0.79;0.845)	0.560	0.560	0.560

In Table 7, using the calculation for Choquet integral just above, the performance of alternative plans are obtained. The defuzzified overall values of alternative plans using generalized Choquet integral are also given in the same table. For example, the value (1.134) of “A1 and overall plan alternative value” is obtained in that way (equation (11)):

\frac{0.318 + 0.616 + 1.775 + 1.827}{4} = 1.134

In addition, when simplified values of the main criteria are taken into consideration, store plan “A1” takes first place according to the “Store presentation area features (C1)”, “Store retail area features (C2)”, and “Store–customer relationship (C3)” criteria.

According to the results in Tables 7 and 8, the defuzzified overall values of alternative plans using generalized Choquet integral are obtained as 1.134, 0.899, and 0.440. This means that the ranking order from the best to the worst is A1, A2, and A3.

Table 8.

Results of the application using generalized Choquet integral.

	Weights	Normalized values
A1	1.1339	45.87%
A2	0.8987	36.35%
A3	0.4396	17.78%

The results of Choquet integral and the comparison with the results of fuzzy AHP and fuzzy ANP are given in Table 9. According to the results, the ranking is obtained as A1 > A2 > A3 in all three methodologies. Given these results, it is fair to say that selecting Alternative A1 is the most reasonable outcome, followed by the others.

Table 9.

Comparison of the results with fuzzy AHP and fuzzy ANP.

	Weights			Normalized Values
	Choquet integral	Fuzzy AHP	Fuzzy ANP	Choquet integral	Fuzzy AHP	Fuzzy ANP
A1	1.1339	2.3766	1.0000	45.87%	69.34%	65.23%
A2	0.8987	0.7892	0.3548	36.35%	23.02%	23.14%
A3	0.4396	0.2618	0.1782	17.78%	7.64%	11.62%

The reason of the variations in the weights and normalized values (Table 9) can be thought of as at fuzzy AHP and fuzzy ANP calculation steps, triangular fuzzy numbers were used and the interactions between main criteria, subcriteria, and alternatives were not entirely taken into account (Ozdemir and Ozdemir, 2017).

The impact of the interdependency among main criteria and subcriteria, the nonlinear relationship among them, the environmental uncertainties and the range computations in Choquet integral, also using trapezoidal fuzzy numbers (at Choquet integral calculation steps) instead of triangular fuzzy numbers (at Fuzzy AHP and Fuzzy ANP calculation steps) are the reasons of inconsistency of weights that can be seen in Table 9. In addition, the proposed methodology (Choquet integral) has an ability of evaluating design process information from internal and external environments.

Conclusion

In the previous study (Ozdemir and Ozdemir, 2017), multi-criteria decision making techniques, fuzzy AHP and fuzzy ANP methods were used for the evaluation of store plan alternatives that produced with shape grammar with triangular fuzzy numbers. In this study, as an extension of fuzzy AHP and fuzzy ANP results, another MCDM technique, the generalized Choquet integral method using trapezoidal fuzzy numbers, is used for evaluation with the same data and the result of this method is compared with fuzzy AHP and fuzzy ANP results.

As a result of evaluation process, this MCDM method, generalized Choquet integral, has determined the most suitable result as A1. The ranking of the other alternatives are A2 > A3. According to the results, the ranking is obtained as A1 > A2 > A3 in all three methodologies with different weights and normalized values.

The reason of this difference can be thought of as at ANP and AHP calculation step, triangular fuzzy numbers were used and the interactions between main criteria, subcriteria, and alternatives are not entirely take into account especially at fuzzy AHP steps. On the other hand, a Choquet integral methodology considers interactivity among main criteria and subcriteria. When interactions among criteria exist, Choquet integral is proved to be an adequate aggregation operator by taking into account the interactions (Shieh et al., 2009). Also using trapezoidal fuzzy numbers instead of triangular fuzzy numbers and the range computations using integral can enable better results for daily usage. In addition, the proposed methodology has an ability of evaluating design process information from internal and external environments. The main advantage of the proposed model is to indicate the impact of this interactivity using trapezoidal fuzzy numbers. The main contribution of this paper is to determine the interdependency, the nonlinear relationship and the environmental uncertainties while prioritizing store plan alternatives.

As in the previous study (Ozdemir and Ozdemir, 2017), the general limitation of the proposed model is the costly and exhausting information requested from experts (approx. 90 pairwise comparisons and also tolerance zones per one expert). Other limitations of the model are the preferences of the expert including uncertainty and conflicts and there is often needed more than one expert to make decisions.

As regards future research, plan alternatives produced with other computer-aided architectural design method—cellular automata—and also hybrid methods could be evaluated by these MCDM techniques. On the other hand, the problem could be solved by previous MCDM techniques with trapezoidal fuzzy numbers and more solutions compared for plan alternatives evaluation processes. Also, intelligent software to calculate solutions automatically could be developed.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Yavuz Ozdemir was born in İstanbul (Bakirkoy) in 1985. He graduated from Vefa High School, and received his BSc, MSc, and PhD from Yildiz Technical University, in the Department of Industrial Engineering. He had gained his PhD degree about operations research issues (linear programming, heuristics, genetic algorithm, and management of irregular cases). He also works at Yildiz Technical University. He has several academic and administrative duties. His research areas are transportation, operations research issues as optimization, heuristics, and multicriteria decision making techniques. He has presented several national and international papers.

Sahika Ozdemir was born in Trabzon (Vakfikebir) in 1985. She graduated from Hüseyin Yildiz High School. She received her BSc from İstanbul Technical University, in the Department of Interior Architecture and her MSc from İstanbul Technical University, in the Department of Computational Architectural Design. She is now a PhD candidate in Yildiz Technical University, in the Department of Architectural Design. She also works as a lecturer at İstanbul Sabahattin Zaim University. Her research areas are interior architecture, computational architectural design and store design.

Appendix 1

References

Chen

Tzeng

(2001) Using fuzzy integral for evaluating subjectively perceived travel costs in a traffic assignment model. European Journal of Operational Research 130: 653–664.

Chiou

Tzeng

Cheng

(2005) Evaluating sustainable fishing development strategies using fuzzy MCDM approach. Omega 33: 223–234.

Choo

Schoner

Wedley

(1999) Interpretation of criteria weights in multicriteria decision making. Computers & Industrial Engineering 37(3): 527–541.

Delgado

Herrera

Herrera-Viedma

et al. (1998) Combining numerical and linguistic information in group decision making. Information Sciences 107: 177–194.

Demirel

Cetin Demirel

Kahraman

(2010) Multi-criteria warehouse location selection using Choquet integral. Expert Systems with Applications 37: 3943–3952.

Hsieh

Tzeng

(2004) Fuzzy MCDM approach for planning and design tenders selection in public office buildings. International Journal of Project Management 22: 573–584.

Kahraman

Cebeci

Ruan

(2004) Multi-attribute comparison of catering service companies using fuzzy AHP: The case of Turkey. International Journal of Production Economics 87(2): 171–184.

Kong W, Qu H and Zhang Q (2007) A fuzzy Choquet integral approach to evaluate the capability of supplier. In: International conference of wireless communications networking and mobile computing WICON, Shanghai, China, 21–25 September 2007, pp.4740–4743. Shanghai: IEEE Conference Publications.

Kuo

Chen

(2006) Selection of mobile value-added services for system operators using fuzzy synthetic evaluation. Expert Systems with Applications 30: 612–620.

10.

Meyer

Roubens

(2006) On the use of the Choquet integral with fuzzy numbers in multiple criteria decision aiding. Fuzzy Sets and Systems 157: 927–938.

11.

Mikhailov

(2003) Deriving priorities from fuzzy pairwise comparison judgments. Fuzzy Sets and Systems 134: 365–385.

12.

Mikhailov

Tsvetinov

(2004) Evaluation of services using a fuzzy analytic hierarchy process. Applied Soft Computing 5(1): 23–33.

13.

Ozdemir S (2014) Bicim grameri ile magaza tasarimina yonelik bir model onerisi. MSc Thesis, Istanbul Technical University Graduate School of Natural and Applied Sciences, Turkey.

14.

Ozdemir

(2017) Prioritizing store plan alternatives produced with shape grammar using multi-criteria decision-making techniques. Environment and Planning B: Urban Analytics and City Science. . Epub ahead of print 06 January 2017. DOI: 10.1177/0265813516686566.

15.

Promentilla

MAB

Furuichi

Ishii

et al. (2008) A fuzzy analytic network process for multi-criteria evaluation of contaminated site remedial countermeasures. Journal of Environmental Management 88: 479–495.

16.

Rodríguez

Ortega

Concepción

(2013) A method for the selection of customized equipment suppliers. Expert Systems with Applications 40(4): 1170–1176.

17.

Saaty

(1980) The Analytic Hierarchy Process, New York: McGraw Hill.

18.

Schoner

Wedley

(1989) Ambiguous criteria weights in AHP: Consequences and solutions. Decision Sciences 20(3): 462–475.

19.

Shieh

Liu

(2009) Applying a complexity based Choquet integral to evaluate student’s performance. Expert Systems with Applications 36: 5100–5106.

20.

Tsai

(2006) The evaluation of service quality using generalized Choquet integral. Inform Sciences 176: 640–663.

21.

Yayla

Yildiz

(2013) Generalised Choquet integral algorithm for subcontractor selection in the textile industry – A case study for Turkey. Fibres & Textiles in Eastern Europe 21: 16–21.

22.

Zadeh

(1965) Fuzzy sets. Information and Control 8: 338–353.