Evaluation of trapezoidal fuzzy numbers on AHP based solution of multi-objective programming problems

Abstract

Multi-criteria decision making method is used in such cases having multiple targets. Multi-objective linear programming problems, which deal with uncertain measurements for both objectives and constraints, are solved by fuzzy multi-objective linear programming methods. Analytic hierarchy process (AHP), one of the multi-criteria decision-making methods based on weighting of objectives, is designed to solve complex problems. In this study, multiple criteria decision making problems and decision makers’ opinions on these problems as well as their solution processes are discussed. Firstly, a fuzzy multi objective linear programming problem is solved using the approaches of Zimmerman and Hybrid. In the literature, these methods are applied using triangular fuzzy numbers based on decision makers’ opinions only in cases where the objective function is minimization. In this study, as an alternative to the triangular fuzzy numbers, trapezoidal fuzzy numbers are defined, and both minimization and maximization of objective functions are examined. Finally, a solution algorithm for a fuzzy multi-objective linear programming problem is proposed under some given conditions. The algorithm is applied to a sample problem that involves modeling of a congress organizing which aims to place attendees to accommodations under specific objectives and constraints. In the evaluation phase of the algorithm, targets are weighted by using triangular and trapezoidal fuzzy numbers. At the end, the results of the proposed algorithm are compared with the results of other methods in the literature.

Keywords

Analytic hierarchy process fuzzy multi-objective linear programming trapezoidal fuzzy numbers

1 Introduction

Multi-criteria decision making method is one of the appropriate optimization techniques to overcome the complex decision-making problems and it is used in cases with multiple objectives. For a multi-criteria decision making problem which has more than one target, the targets can be conflicting. Therefore, achieving the best value for all targets at the same time is not always possible. Since the targets and constraints are defined in the form of linear equalities or inequalities, multi-criteria decision making problems deal with the optimization of linear functions. Nevertheless, in general, parameter values related to the defined functions cannot be determined with certainty by decision makers. This situation can be handled by the fuzzy multi-criteria linear programming models, where all of the objective functions of a problem may not be at the same level of importance or at the same priorities. In these cases, by weighting objective functions using AHP method, decision making process is completed [22].

In this study, multi-criteria decision making problems and decision makers’ opinions for these problems have been handled. At first, fuzzy linear programming problems were solved by Zimmerman approach, which exists in literature. Then, decision makers’ assessments were converted into triangular fuzzy numbers and relative weights were obtained from these numbers. By using these attained weights, the same problems were modeled again. As an alternative to this solution, located in the literature, decision makers’ opinions were defined as fuzzy trapezoidal numbers. By weighting triangular and trapezoidal fuzzy numbers as two different formats, the problems were solved with Hybrid approach. At the next stage, as an alternative to the cases in the literature where the target was only the minimization of objective functions, cases which include some minimization and maximization objective functions in the same problem were discussed. Finally, the comparison of the results of three methods (Zimmerman approach, hybrid approach which uses the weights acquired from triangular fuzzy numbers, hybrid approach which uses the weights acquired from trapezoidal fuzzy numbers) has been made. The relevant literature on fuzzy multi objective programming and AHP is discussed below.

In 1990, Dombi defined different kinds of membership functions and constructed fundamental features and mathematical forms for these functions [10]. Chang introduced a new approach for handling fuzzy AHP, with the use of triangular fuzzy numbers for pairwise comparison scale of fuzzy AHP [5]. Chen and Wang proposed a heuristic method to calibrate the fuzzy exponent iteratively, and presented a hybrid learning algorithm for refining the system parameters [7]. Jana and Roy worked on the solutions of multi-objective fuzzy linear programming problems, as well as the solutions of transportation problems. Their study consisted of two stages. In the first stage, they defined the fuzzy objective coefficients and the fuzzy constraints functions by using fuzzy triangle numbers. Then in the second stage, the transportation problem has been handled in the fuzzy framework [11]. Kaya and Kahraman have used the VIKOR-AHP approach to choose among alternative forestation areas in Istanbul [13]. Rahman and Ahsan implemented a comprehensive fuzzy AHP analysis to evaluate suppliers and to determine the best supplier [18]. Javanbarg et al. converted a non-linear optimization model to a prioritized fuzzy problem and applied particle swarm optimization for its solution [12]. Shaw et al. presented an integrated approach for selecting the appropriate supplier in the supply chain, addressing the carbon emission issue, using fuzzy-AHP and fuzzy multi-objective linear programming [21]. Calabrese et al. proposed a model for intellectual capital evaluation by integrating fuzzy logic and AHP [3]. Song et al. employed a fuzzy AHP approach based on triangular and trapezoidal fuzzy numbers to evaluate self-ignition risks of spontaneous combustion of coal stockpiles [23]. Chan et al. introduced a comprehensive method that integrates the Life-Cycle Assessment (LCA) and Environmental Management Accounting (EMA) concepts, fuzzy logic and AHP, to measure the environmental and organizational performance of different designs, and they proposed a screening model to help designers reduce their reliance on LCA [6]. Mosadeghi et al. compared the results of two quantitative techniques (AHP and fuzzy AHP) in defining the extent of land-use zones at a large scale urban planning scenario [17]. Tadic et al. have used triangular fuzzy numbers to modelling the weights of each pair of the considered criteria. They calculated weights vector of criteria by the fuzzy AHP and applied fuzzy Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) to rank quality goals [24]. In another paper of Tadic et al. applied fuzzy AHP to determine the relative importance of business process and applied fuzzy TOPSIS to rank the organizational resilience factors. They introduced a modified fuzzy decision matrix [25].

The organization of the paper is as follows: In Section 2, fuzzy multi-criteria decision making problems with more than one objective are discussed. Otherwise in order to handle the problem that multi objectives have different weights which are uncertain, fuzzy AHP method is introduced. In Section 3, an algorithm based on trapezoidal fuzzy numbers is given for weighting. In Section 4, an appropriate application to the purpose of our study has been provided. In application, for a congress which will be held in a city, an optimum way to distribute participants to hotels is planned under specific objectives and constraints. Objective priorities are determined by decision makers. These priorities are transformed into fuzzy weights by using a MATLAB code, which is constructed according to a designed weight algorithm. Thereafter, a model is implemented on the solution of the application, and operated on WINQSB to obtain the results, which is discussed in the last section.

2 Fuzzy multi-criteria decision making

Multi-criteria decision making method is one of the appropriate optimization techniques to overcome the complex decision-making problems and it is used in cases with multi objectives. The model relative to the solution of multi-criteria decision making problems is established to determine the best objective value that satisfies the decision maker in the appropriate solution space. The general mathematical structure of multi-criteria decision-making models is, $P_{1} {\begin{matrix} max z_{1} = c \underline{x} = f_{1} \\ ⋮ \\ max z_{j} = c \underline{x} = f_{j} \\ g_{k} (x) \leq 0 (k = 1 \dots K) \\ (j = 1, \dots, J) \underline{x} \geq 0 \end{matrix}$ (1)

where,

f_j: j. objective function,

g_k: k constraint function,

J: The number of objective functions,

K: The number of constraint functions,

$\underline{x}$ : The decision variables vector.

In multi criteria linear programming problems, when value of the function parameters on objects and constraints can not be measured precisely by experts and decision makers, fuzzy multi criteria linear programming methods are used to solve the problems. The state of fuzziness in a linear programming problem occurs in three ways. The first one takes place when the decision maker certainly knows the objective function, but the right-hand side values are fuzzy. The second case occurs when the right-hand side values are known by the decision maker, but the parameters of the objective function are fuzzy. The third case takes place when the whole model is fuzzy, that is, the decision maker does not know the parameters of the objective function and the right-hand side values with certainty [28].

The solution approaches of Verdegay and Werners are used for linear programming problems where right-hand side values are fuzzy. Verdegay proposed solution approaches for the cases where only the right-hand side values are fuzzy. Nevertheless, Werners stated that the objective function should also be fuzzy because of the fuzzy constraints. The levels of demand for both the right-hand side values and the objective function are determined by interacting with the decision maker. It is stated in Zimmerman’s approach that the objective and the right-hand side values with their corresponding tolerances are defined by the decision maker initially [14, 20].

Fuzzy linear programming was first used by Zimmerman as a decision model [9]. For the models of fuzzy objective and fuzzy constrained linear programming, Zimmerman suggested that the decision maker could determine the amount of tolerance that he/she aims for the objective function, prior to the solution. The fuzzy linear programming problem identified by Zimmerman can be expressed as; $\begin{matrix} c^{T} x \tilde{\geq} b_{0} \\ (A_{x})_{k} \tilde{\leq} b_{k} \forall k k = \bar{1, K} \end{matrix}$ (2) $x \geq 0$

The signs ‘ $\tilde{\geq}$ ’ and ‘ $\tilde{\leq}$ ’ represent fuzzy inequalities. That is, the first inequality expresses either c^Tx is more than or close to b₀, and the second inequality expresses either Ax is less than or near b.

The symmetric fuzzy linear programming problems are remodeled, and λ is aimed to be maximized in this new model [4, 14], which is shown in Equation (3).

$\begin{matrix} Max λ \\ c^{T} x \geq b_{0} - (1 - λ) p_{0} \\ (Ax)_{k} \leq b_{k} + (1 - λ) p_{k} \\ x \geq 0, λ \in [0, 1] \end{matrix}$ (3)

Fuzzy AHP, which is modeled with P₁, is one of the processes in the literature for the solution of multi-criteria decision-making problems.

Analytic hierarchy process (AHP) determines the relative importance of each criterion determined by the decision makers [2]. Then, based on each criterion, a choice among the alternatives has to be made. The scale of 1–9 advised by Saaty is used when comparing criterions in the AHP method [19]. Apart from this, there are also other importance scales, such as 1–5, 1–7, 1–15 and 1–20. However, the best results are obtained from 1–9 importance scale (for details, see in [8, 12]).

Although AHP’s aim is to reveal the knowledge of decision makers, conventional AHP still do not reflect the human thought accurately. On real world problems, instead of using exact numbers that represent the decision makers’ opinions, verbal assessment is more realistic and appropriate. Many researchers have been interested in Saaty’s AHP method since it provides more precise definitions than the traditional AHP techniques. Fuzzy AHP method can be considered as an improved version of AHP which is used for modeling unstructured problems from various fields [12 , 26].

In the classic AHP method, the relative importance of each criterion are determined using a 1–9 significance scale, whereas in the fuzzy AHP method, criteria comparison is done by using fuzzy numbers and linguistic variables. Decision makers generally prefer triangular fuzzy numbers in paired comparisons. Fuzzy importance scales can vary according to research interests. Linguistic value for importance scale to be used, are given in the literature. Linguistic scales for importance, triangular fuzzy numbers and triangular fuzzy reciprocal numbers are given (for details, see in [1 , 21]). In this paper, the fuzzy rating of each decision maker can be described by using four linguistic expressions which are modeled by triangular fuzzy numbers:

equally important: (1, 1, 1)

moderately important: (1, 2, 3)

important: (2, 3, 4)

very strongly important: (3, 4, 5).

3 An algorithm for weighting of objective functions by using trapezoidal fuzzy numbers

In the early studies on classical AHP method, certain importance levels were used. In subsequent studies, triangle fuzzy numbers, which are the expansions of the exact numbers, were used to define the importance levels. In this study, importance levels were obtained by using trapezoidal fuzzy numbers, an expansion of triangular fuzzy numbers, and the change in the objective function value was examined. When the trapezoidal fuzzy numbers used in weight calculation, the following algorithm steps are performed for the Hybrid approach:

Step 1: The maximum and minimum values of each objective function, and the differences between these values are calculated.

Step 2: If the objective is minimization, the membership function is as follows:

$\begin{matrix} μ_{zj} (x) \\ = {\begin{matrix} 1, & Z_{j} (x) \leq Z_{j}^{min} \\ \frac{[Z_{j}^{max} - Z_{j} (x)]}{[Z_{j}^{max} - Z_{j}^{min}]}, & Z_{j}^{min} \leq Z_{j} (x) \leq Z_{j}^{max} \\ 0, & Z_{j} (x) \geq Z_{j}^{max} \end{matrix} \end{matrix}$ (4)

If the objective function is maximization, the membership function is as follows:

$\begin{matrix} μ_{Zj} (x) \\ = {\begin{matrix} 1, & Z_{j} (x) \leq Z_{j}^{min} \\ \frac{[Z_{j} (x) - Z_{j}^{min}]}{[Z_{j}^{max} - Z_{j}^{min}]}, & Z_{j}^{min} \leq Z_{j} (x) \leq Z_{j}^{max} \\ 0, & Z_{j} (x) \geq Z_{j}^{max} \end{matrix} \end{matrix}$ (5)

The linear membership function for the fuzzy constraints is given by Equation (6):

$\begin{matrix} μ_{gt} (x) \\ = {\begin{matrix} 1, & g_{t} (x) \leq b_{t} \\ [1 - \frac{{g_{t} (x) - b_{t}}}{d_{t}}], & b_{t} \leq g_{t} (x) \leq b_{t} + d_{t} \\ 0, & b_{t} + d_{t} \leq g_{t} (x) \end{matrix} \end{matrix}$ (6) for all fuzzy parameters t = 1, … T. The interpretation of d_t is the tolerance value.

Step 3:n being the number of criteria and m being the number of decision makers involved, comparison values (a_mnp) relative to each criterion are determined by the decision makers as in Table 1.

A trapezoidal fuzzy number consists of four parameters indicated as a¹, a², a³, a⁴ and is expressed as follows: $a_{mnp} = (a_{mnp}^{1} a_{mnp}^{2} a_{mnp}^{3} a_{mnp}^{4})$ (7)

a_mnp: The importance value of m^th criteria corresponding to n^th criteria, according to p^th decision maker

m, n: number of criteria (m : 1, …, L, n : 1, …, L)

p: number of decision maker (p: 1, ... , P)

l: number of criteria (l = 1, … , L) (m ≠ n)

The trapezoidal membership function, shown in Fig. 1, is expressed as follows: $μ_{\tilde{A}} = {\begin{matrix} \frac{x - a^{1}}{a^{2} - a^{1}}, & a^{1} < x < a^{2} \\ 1, & a^{2} \leq x \leq a^{3} \\ \frac{a^{4} - x}{a^{4} - a^{3}}, & a^{3} < x < a^{4} \\ 0, & x \geq a^{4} and x \leq a^{1} \end{matrix}$ (8)

Step 4: New trapezoidal fuzzy numbers are obtained with pairwise comparisons of L criteria shown in Table 1. For the comparison of C_m and C_n criteria, $\begin{matrix} a_{mnp} & = & (a_{mnp}^{1} a_{mnp}^{2} a_{mnp}^{3} a_{mnp}^{4}), where \\ a_{mn}^{1} & = & {(a_{mn 1}^{1} * a_{mn 2}^{1} * \dots * a_{mnp}^{1})}^{\frac{1}{p}} \\ a_{mn}^{2} & = & {(a_{mn 1}^{2} * a_{mn 2}^{2} * \dots * a_{mnp}^{2})}^{\frac{1}{p}} \end{matrix}$ (9) $\begin{matrix} a_{mn}^{3} & = & {(a_{mn 1}^{3} * a_{mn 2}^{3} * \dots * a_{mnp}^{3})}^{\frac{1}{p}} \\ a_{mn}^{4} & = & {(a_{mn 1}^{4} * a_{mn 2}^{4} * \dots * a_{mnp}^{4})}^{\frac{1}{p}}, and \\ A_{mn} & = & [a_{mn}^{1} a_{mn}^{2} a_{mn}^{3} a_{mn}^{4}] \end{matrix}$ (10)

The new trapezoidal fuzzy numbers, which express the decision makers’ opinions, are obtained by repeating the same process L times for each paired comparison, using geometric mean [27].

Step 5: The values of membership degree for each criterion (F_l) are as follows [5 , 29]: $\begin{matrix} F_{1} = \sum_{n = 1}^{L} A_{1 n} \otimes [M_{mn}]^{- 1} \\ ⋮ \\ F_{L} = \sum_{n = 1}^{L} A_{Ln} \otimes [M_{mn}]^{- 1} \\ (m = 1, 2, \dots, L, n = 1, 2, \dots, L) \end{matrix}$ (11)

The sum of the fuzzy numbers (A_mn) given in Equation (10) are obtained by $M_{mn} = \sum_{m = 1}^{L} \sum_{n = 1}^{L} A_{mn} = (\begin{matrix} m_{mn}^{1} & m_{mn}^{2} & m_{mn}^{3} & m_{mn}^{4} \end{matrix})$ (12)

The inverse of the resulting sum is computed as follows: ${[M_{mn}]}^{- 1} = (\begin{matrix} \frac{1}{m_{mn}^{4}} & \frac{1}{m_{mn}^{3}} & \frac{1}{m_{mn}^{2}} & \frac{1}{m_{mn}^{1}} \end{matrix})$ (13)

Step 6: The values of membership are compared based on the calculated F_l (l = 1, …, L) in Step 5. Comparison method for two triangular fuzzy numbers, in the literature [16], has been developed for trapezoidal fuzzy number in this study, as follows:

When $f_{1}^{1} \geq f_{2}^{1}, f_{1}^{2} \geq f_{2}^{2}, f_{1}^{3} \geq f_{2}^{3}, f_{1}^{4} \geq f_{2}^{4}$ asin Fig. 2, the trapezoidal fuzzy number $F_{1} (f_{1}^{1}, f_{1}^{2}, f_{1}^{3}, f_{1}^{4})$ is greater than the trapezoidal fuzzy number $F_{2} (f_{2}^{1}, f_{2}^{2}, f_{2}^{3}, f_{2}^{4})$ , and the value of membership (μ_d) is expressed as V (F₁ ≥ F₂) = μ_d = 1.

As in Fig. 3, when $f_{1}^{1} \geq f_{2}^{4}$ , the value of membership (μ_d) is expressed as V (F₁ ≥ F₂) = 0.

In other cases, the value of membership (μ_d) is expressed as; $V (F_{2} \geq F_{1}) = μ_{d} = \frac{f_{1}^{1} - f_{2}^{4}}{(f_{2}^{3} - f_{2}^{4}) - (f_{1}^{2} - f_{1}^{1})}$ (14)

Step 7: Each trapezoidal fuzzy number F_l is compared with all other F_l trapezoidal fuzzy numbers. Then, the minimum F_l is obtained and represented by the weight vector d (F_l) in Equation (15) [5, 15, 29].

$\begin{matrix} d (F_{1}) & = & Min {V (F_{1} \geq F_{2}), \dots, V (F_{1} \geq F_{L})} \\ ⋮ \\ d (F_{L}) & = & Min {V (F_{l} \geq F_{1}), \dots, V (F_{L} \geq F_{L - 1})} \end{matrix}$ (15)

Step 8: The weight vector is obtained by using d (F_l) as follows [5 , 29]: $W^{'} = {(ω_{1}, ω_{2}, ..., ω_{L})}^{T}$ (16) $where w_{l} = d (F_{l}) .$

Step 9: By utilizing w_l obtained by trapezoidal fuzzy numbers, corresponding to opinion of decision makers, the linear programming problem is developed as; $\begin{matrix} Max \sum_{l = 1}^{L} w_{l} λ_{l} + \sum_{t = 1}^{T} β_{t} γ_{t} \\ λ_{l} \leq μ_{Z_{l}} (x), γ_{t} \leq μ_{g_{t}} (x), g_{k} (x) \leq b_{k}, \\ \sum_{l = 1}^{L} w_{l} + \sum_{t = 1}^{T} β_{t} = 1 \end{matrix}$ (17) $\begin{matrix} λ_{l,} γ_{t} \in [0, 1], w_{l}, β_{t} \geq 0, \forall x \in Z^{+}, \\ l = 1, \dots, L, k = 1, 2, \dots, K, t = 1, \dots, T, \end{matrix}$

In Equation (17), w_l, β_t are the weights coefficients that present the relative importance among the fuzzy goals and fuzzy constraints. γ_t is a fuzzy constraint that obtained by using by Equation (6).

Step 10: By solving the linear programming problem at Step 9, the values of the decision variables x and the objective function value are determined.

4 Application

A scientific convention will be held in Trabzon, Turkey. It is believed that about 500 people will attend this congress, and it is assumed that it can vary from 475 to 525. For all seven hotels where the congress participants can accommodate, the organizing company has determined the room prices, the distances from the convention center, the distances from the city center, customer satisfaction, customer comfort, and the capacities of the hotels. The aim is to find an optimum way to distribute participants to 7 hotels in accordance with the constraints. Quantitative information regarding to the specified criteria is presented in Table 2.

The goal of the organizing company is to minimize the total room price of the hotels, total distances to the convention center and total distances from the city center, while achieving the maximum customer satisfaction and hotel comfort.

According to the hotel information given in Table 2, the problem is modeled and capacity constraint of the hotels are given as: $P_{2} : Z_{1 (\min)} = 242 x_{1} + 250 x_{2} + \dots + 90 x_{7}$ (18) $Z_{2 (\min)} = 8.5 x_{1} + 19 x_{2} + \dots + 0.10 x_{7}$ (19) $Z_{3 (\min)} = 0.36 x_{1} + 22.6 x_{2} + \dots + 6.9 x_{7}$ (20) $Z_{4 (\max)} = 82 x_{1} + 84 x_{2} + \dots + 79 x_{7}$ (21) $Z_{5 (\max)} = 95 x_{1} + 85 x_{2} + \dots + 80 x_{7}$ (22) $x_{1} + x_{2} + \dots + x_{7} = 500$ (23) $\begin{matrix} x_{1} \leq 157, x_{2} \leq 200, x_{3} \leq 66, x_{4} \leq 69, \\ x_{5} \leq 120, x_{6} \leq 88, x_{7} \leq 108 \end{matrix}$ (24) $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7} \in Z^{+} .$

The number of participants who will be accommodated in hotel i (i = 1, …, 7),

Objective function regarding to the room prices (18),

Objective function regarding to the total distances from the convention center (19),

Objective function regarding to total distances from the city center (20),

Objective function regarding to the customer satisfaction (21),

Objective function regarding to the hotel comfort (22),

which is given by Equation (23) is number of estimated people number, will attend the congress,

which is given by Equation (24) is capacity constraint of the hotels.

The maximum and minimum values for each objective function were determined under the constraints of the model by using WinQSB software. Solutions for each of 5 objective functions are given in Table 3.

If the problem is modeled with Zimmerman approach using the results in Table 3, the following mathematical model (P₃) is obtained: $\begin{matrix} P_{3} : & Max λ \\ λ \leq \frac{115964 - (242 x_{1} + 250 x_{2} + \dots + 90 x_{7})}{49831} \\ λ \leq \frac{7816.4 - (8.5 x_{1} + 19 x_{2} + \dots + 0.10 x_{7})}{4733.1} \\ λ \leq \frac{8680.4 - (0.36 x_{1} + 22.6 x_{2} + \dots 6.9 x_{7})}{6910.28} \\ λ \leq \frac{(82 x_{1} + 84 x_{2} + \dots + 79 x_{7}) - 33049}{9717} \\ λ \leq \frac{(95 x_{1} + 85 x_{2} + \dots + 80 x_{7}) - 34770}{10285} \\ λ \leq \frac{525 - (x_{1} + x_{2} + \dots x_{7})}{25} \\ λ \leq \frac{(x_{1} + x_{2} + \dots + x_{7}) - 475}{25} \\ x_{1} \leq 157, x_{2} \leq 200, x_{3} \leq 66, x_{4} \leq 69 \\ x_{5} \leq 120, x_{6} \leq 88, x_{7} \leq 108 \\ x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7} \in Z^{+} \end{matrix}$

The value of the objective function is obtained as λ = 0, 5757. The values of the decision variables are found as x₁ = 157, x₂ = 19, x₃ = 66, x₄ = 69, x₅ = 3, x₆ = 88, x₇ = 108, and after replacing these values in P₂, the values of the objective functions are calculated as Z₁ = 87274, Z₂ = 4877.9, Z₃ = 4689.3, Z₄ = 38648, Z₅ = 41070.

It has been used triangular fuzzy numbers in the literature in order to compare the utility of results obtained by trapezoidal fuzzy numbers. The weight based problem formed by using triangle fuzzy numbers is remodeled with the hybrid approach by using the results of Table 3. The obtained model is given as P₄. $\begin{matrix} P_{4} : & Max 0.2720 λ_{1} + 0.2574 λ_{2} + \dots + 0.0013 γ_{1} \\ λ_{1} \leq \frac{115964 - (242 x_{1} + 250 x_{2} + \dots + 90 x_{7})}{49831} \\ λ_{2} \leq \frac{7816.4 - (8.5 x_{1} + 19 x_{2} + \dots + 0.10 x_{7})}{4733.1} \\ λ_{3} \leq \frac{8680.4 - (0.36 x_{1} + 22.6 x_{2} + \dots + 6.9 x_{7})}{6910.28} \\ λ_{4} \leq \frac{(82 x_{1} + 84 x_{2} + \dots + 79 x_{7}) - 33049}{9717} \\ λ_{5} \leq \frac{(95 x_{1} + 85 x_{2} + \dots + 80 x_{7}) - 34770}{10285} \\ γ_{1} \leq \frac{525 - (x_{1} + x_{2} + \dots + x_{7})}{25} \\ γ_{1} \leq \frac{(x_{1} + x_{2} + \dots + x_{7}) - 475}{25} \\ x_{1} \leq 157, x_{2} \leq 200, x_{3} \leq 66, x_{4} \leq 69 \\ x_{5} \leq 120, x_{6} \leq 88, x_{7} \leq 108 \\ x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7} \in Z^{+} \end{matrix}$

When the model in P₄ is processed by using WinQSB software, the following results are gathered:

λ = 0, 7258, λ₁ = 0.7187, λ₂ = 1, λ₃ = 0.9224, λ₄ = 0.2325, λ₅ = 0.3797, γ₁ = 0,

x₁ = 157, x₂ = 0, x₃ = 0, x₄ = 2, x₅ = 120, x₆ = 88, x₇ = 108.

After replacing these values in P₂, the values of the objective functions are calculated as Z₁ = 104348, Z₂ = 3083.3, Z₃ = 2306.1, Z₄ = 35308, Z₅ = 38675.

In this paper, the fuzzy rating of each decision maker can be described by using four linguistic expressions, which are modeled by trapezoidal fuzzy numbers as follows:

equally important: (1, 1, 1, 1)

moderately important: (1, 1.6, 2.4, 3)

important: (2, 2.6, 3.4, 4)

very strongly important: (3, 3.6, 4.4, 5)

Table 4 is obtained by taking the geometric mean of decision maker opinions (Using Equation (9)). By the implementation of the proposed algorithm weights can be obtained as follows,

$\begin{matrix} M_{mn} & = & \sum_{m = 1}^{6} \sum_{n = 1}^{6} A_{mn} \\ = & (\begin{matrix} 35.048 & 45.209 & 59.276 & 71.126 \end{matrix}) \\ {[M_{mn}]}^{- 1} & = & (\begin{matrix} 0.014 & 0.016 & 0.022 & 0.028 \end{matrix}) \end{matrix}$

By using Equation (11), the values of membership degree for each criterion is determined as below;

$\begin{matrix} \sum_{n = 1}^{6} A_{1 n} = (8.668 11.815 15.917 18.964), \\ ⋮ \\ \sum_{n = 1}^{6} A_{6 n} = (2.363 2.635 3.235 4.117) \\ F_{1} = (0.121 0.199 0.352 0.541) \\ ⋮ \\ F_{6} = (0.033 0.044 0.071 0.117) \end{matrix}$

By using comparison method for trapezoidal fuzzy numbers, in Step 6 of proposed algorithm; $\begin{matrix} V (F_{1} \geq F_{2}) = 0.542, V (F_{1} \geq F_{3}) = 1, \\ V (F_{1} \geq F_{4}) = 1, V (F_{1} \geq F_{5}) = 1, \\ V (F_{1} \geq F_{6}) = 1; \\ V (F_{2} \geq F_{1}) = 0.457, V (F_{2} \geq F_{3}) = 1, \\ V (F_{2} \geq F_{4}) = 1, V (F_{2} \geq F_{5}) = 1, \\ V (F_{2} \geq F_{6}) = 1; \\ V (F_{3} \geq F_{1}) = 0.194, V (F_{3} \geq F_{2}) = 0.225, \\ V (F_{3} \geq F_{4}) = 1 V (F_{3} \geq F_{5}) = 1, \\ V (F_{3} \geq F_{6}) = 1; \\ V (F_{4} \geq F_{1}) = 0, V (F_{4} \geq F_{2}) = 0, \\ V (F_{4} \geq F_{3}) = 0.181, V (F_{4} \geq F_{5}) = 1, \\ V (F_{4} \geq F_{6}) = 1; \\ V (F_{5} \geq F_{1}) = 0, V (F_{5} \geq F_{2}) = 0, \\ V (F_{5} \geq F_{3}) = 0, V (F_{5} \geq F_{4}) = 0.205, \\ V (F_{5} \geq F_{6}) = 1; \\ V (F_{6} \geq F_{1}) = 0, V (F_{6} \geq F_{2}) = 0, \\ V (F_{6} \geq F_{3}) = 0, V (F_{6} \geq F_{4}) = 0, \\ V (F_{6} \geq F_{5}) = 0 . \end{matrix}$

The weight of criteria is obtained by using Equation (15); $\begin{matrix} d (F_{1}) = 0.543, d (F_{2}) = 0.457, d (F_{3}) = 0.194 \\ d (F_{4}) = 0, d (F_{5}) = 0, d (F_{6}) = 0 \end{matrix}$

The weight vector is obtained by using Equation (16); $\begin{matrix} W^{'} & = & {(d (F_{1}) d (F_{2}) d (F_{3}) d (F_{4}) d (F_{5}) d (F_{6}))}^{T} \\ = & {(0.543 0.457 0.194 0 0 0)}^{T} \\ = & (0.4544 0.3825 0.1631 0 0 0) \end{matrix}$

The weight based problem formed by using trapezoidal fuzzy numbers is remodeled with the hybrid approach by using the results of Table 3. The obtained model is given as P₅. $\begin{matrix} P_{5} : & Max 0.4544 λ_{1} + 0.3825 λ_{2} + \dots + 0 γ_{1} \\ λ_{1} \leq \frac{115964 - (242 x_{1} + 250 x_{2} + \dots + 90 x_{7})}{49831} \\ λ_{2} \leq \frac{7816.4 - (8.5 x_{1} + 19 x_{2} + \dots + 0.10 x_{7})}{4733.1} \\ λ_{3} \leq \frac{8680.4 - (0.36 x_{1} + 22.6 x_{2} + \dots + 6.9 x_{7})}{6910.28} \\ λ_{4} \leq \frac{(82 x_{1} + 84 x_{2} + \dots + 79 x_{7}) - 33049}{9717} \\ λ_{5} \leq \frac{(95 x_{1} + 85 x_{2} + \dots + 80 x_{7}) - 34770}{10285} \\ γ_{1} \leq \frac{525 - (x_{1} + x_{2} + \dots + x_{7})}{25} \\ x_{1} \leq 157, x_{2} \leq 200, x_{3} \leq 66, x_{4} \leq 69 \\ x_{5} \leq 120, x_{6} \leq 88, x_{7} \leq 108 \\ x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7} \in Z^{+} \end{matrix}$

When the model in P₅ is processed by using WinQSB software, the following results are gathered: $\begin{matrix} λ = 0, 8620, λ_{1} = 0.8384, λ_{2} = 8967, \\ λ_{3} = 0.8464, λ_{4} = 0.1291, λ_{5} = 0.1517, γ_{1} = 0 . \\ x_{1} = 90, x_{2} = 0, x_{3} = 0, x_{4} = 69, x_{5} = 120, \\ x_{6} = 88, x_{7} = 108, \end{matrix}$

After replacing these values in P₂, the values of the objective functions are calculated as Z₁ = 87274, Z₂ = 4877.9, Z₃ = 4689.3, Z₄ = 38648, Z₅ = 41070. The results obtained from the application of different methods are summarized in Table 5.

5 Discussion

In this study, for a scientific conference which will be held in Trabzon, Turkey, around 500 participants were planned to be accommodated in 7 hotels by an organizing company. The aim of the company was to manage this organization in an optimum manner by minimizing the total room price of the hotels, total distances to the convention center and total distances from the city center, while maximizing the customer satisfaction and hotel comfort.

In order to solve this multi-objective linear problem, the method of fuzzy AHP has been employed. The decision makers’ opinions on the comparison of the hotel features have been gathered. The problem was initially solved by Zimmerman approach. Thereafter, the same problem was solved by Hybrid approach by weighting decision makers’ opinions in two different ways: (1) by using triangle fuzzy numbers and (2) by using symmetric trapezoidal fuzzy numbers which is proposed in this paper.

The objective functions values of model P₂, given by (Equations 18–22), are determined using value of decision variable, obtained by solving models P₃, P₄, P₅. And these obtained objective function values are given in Table 5. Values of objective functions, obtained from Zimmerman approach, Hybrid approach with triangular fuzzy numbers and Hybrid approach with trapezoidal fuzzy numbers which is proposed in this paper, are expressed as respectively λ_Z, λ_T and λ_Tr. The final solution is obtained as λ_Z = 0.5757, λ_T = 0.7258 and λ_Tr = 0.8620. It is seen obviously that the proposed method gives better result than the others.

The proposed algorithm can be used as an alternative for the solution of similar multi-criteria decision-making problems. In the later stages of the study, the resolution process can be examined after defining unsymmetrical trapezoidal fuzzy numbers.

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