Interactive intuitionistic fuzzy methods for multilevel programming problems

Accepted Manuscript

Interactive intuitionistic fuzzy methods for multilevel programming problems Xiaoke Zhao, Yue Zheng, Zhongping Wan PII: DOI: Reference:

S0957-4174(16)30618-2 10.1016/j.eswa.2016.10.063 ESWA 10968

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Expert Systems With Applications

Received date: Revised date: Accepted date:

25 July 2016 22 October 2016 31 October 2016

Please cite this article as: Xiaoke Zhao, Yue Zheng, Zhongping Wan, Interactive intuitionistic fuzzy methods for multilevel programming problems, Expert Systems With Applications (2016), doi: 10.1016/j.eswa.2016.10.063

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Highlights • Three interactive intuitionistic fuzzy methods are proposed for MLPPs. • A score function is defined to depict decision makers’ satisfactory degree.

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• Use a new distance function to select a priority solution.

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• A case study and numerical results show that the proposed methods are efficient.

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Interactive intuitionistic fuzzy methods for multilevel

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programming problems

Xiaoke Zhaoa , Yue Zhengb,∗, Zhongping Wana,c

a School of Mathematics and Statistics, Wuhan University, Wuhan 430072, P. R. China

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b School of Management, Huaibei Normal University, Huaibei 235000, P. R. China

c Computational Science Hubei Key Laboratory, Wuhan University, Wuhan, 430072, P. R. China

Abstract

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Multilevel programming problems model a decision-making process with a hierarchy structure. Traditional solution methods including vertex enumeration algorithms

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and penalty function methods are not only inefficient to obtain the solution of the multilevel programming problems, but also lead to a paradox that the follower’s de-

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cision power dominates the leader’s. In this paper, both multilevel programming and intuitionistic fuzzy set are used to model problems in hierarchy expert and intelligent

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systems. We first present a score function to objectively depict the satisfactory degrees of decision makers by virtue of the intuitionistic fuzzy set for solving multilevel programming problems. Then we develop three optimization models and three interactive

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intuitionistic fuzzy methods to consider different satisfactory solutions for the requirements of expert decision makers. Furthermore, a new distance function is proposed to measure the merits of a satisfactory solution. Finally, a case study for cloud computing pricing problems and several numerical examples are given to verify the applicability and the effectiveness of the proposed models and methods.

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Corresponding author. E-mail addresses: [email protected] (X. Zhao), [email protected]

(Z. Wan), [email protected] (Y. Zheng).

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Key words. multilevel programming; interactive intuitionistic fuzzy method; satisfactory solution; decision making

Introduction

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1.

Multilevel programming problems (MLPPs), which are the crucial mathematical optimization problems for describing large decentralized decision problems with multiple interacting decision makers in a hierarchical organization, have been widely applied in industry (Nicholls, 1996), agriculture (Candler et al., 1981), transportation (Suh and Kim, 1992),

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government policy (Amouzegar and Moshirvaziri, 1999), finance (Bard et al., 2000), warfare (Bracken et al., 1977), planning (Hobbs and Nelson, 1992), cloud computing resource allocation (Yeh et al., 2012), single image super-resolution (Li et al., 2014), label propagation (Zoidi et al., 2014), municipal waste system (Saranwong and Likasiri, 2016) and supply

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chain management (Gao et al., 2011; Saranwong and Likasiri, 2016). Decision makers make decisions in sequence from the top level to the bottom level. The upper level has a priority

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over the lower level but it also depends on the reactions of the lower levels. Moreover, each decision maker optimizes his objective function as far as possible.

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MLPPs (Bard and Falk, 1982; Benson, 1989; Hansen et al., 1992; Migdalas et al., 2013; Vicente and Calamai, 1994; Zhang et al., 2015; Zhang et al., 2016), in particular, bilevel

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programming problems (BLPPs) (Bialas and Karwan, 1984; Dempe, 2002; Hansen et al., 1992; Vicente and Calamai, 1994; Zheng et al., 2014) and trilevel programming problems

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(TLPPs) (Lu et al., 2016) have been studied by many authors. Blair (Blair, 1992) noted that MLPPs were NP-hard and the complexity increased significantly when the number of levels was greater than two. There are six main approaches for solving MLPPs: the decent method (Kolstad and Lasdon, 1990), the approach based on Karush-Kuhn-Tucker conditions (Lu et al., 2007), the cutting plane algorithm (Bard, 1984), the penalty function approach (White, 1997), the heuristics method (Hosseini and Kamalabadi, 2014) and the vertex enumeration (Han et al., 2015). 3

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However, Lai (1996) pointed out that the traditional methods led to a paradox that the follower’s decision power dominated the leader’s and proposed a satisfactory solution concept for hierarchical optimization problems. Shih et al. (1996) extended the satisfactory solution concept of Lai (1996) to solve MLPPs. Unfortunately, there is a possibility that their

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methods lead to undesirable solutions because of the inconsistency between fuzzy goals of objective functions and decision variables. To overcome this situation, Sakawa et al. (1998) proposed an interactive fuzzy programming for multilevel linear programming problems with eliminating the fuzzy goals of decision variables. Pramanik and Roy (2007) presented a linear fuzzy goal programming approach for MLPPs to obtain a satisfactory solution, which can

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derive the highest membership degree goal by minimizing negative deviational variables. Arora and Gupta (2009) presented an interactive fuzzy goal programming approach with the characteristics of dynamic programming to solve bilevel programming problems. Dempe (2011) made an comment on the approach of Arora and Gupta (2009) and presented an

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crisp substitute to arrive a satisfactory solution. Zheng et al. (2014) proposed a new balance function and a distance function by considering the overall satisfactory balance between the

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leader and the follower. In a word, the fuzzy methods mentioned above were realized by means of membership degrees to depict decision makers’ satisfactory degrees.

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Unfortunately, there is no objective procedure to determine a membership degree by the fuzzy set theory. To increase objectivity, it is reasonable to use an interval to represent a

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satisfactory degree, i.e. interval-valued fuzzy set theory which is equivalent to intuitionistic fuzzy set (IFS) theory (Dubey et al., 2012). As is well-known, IFS (Atanassov, 1999) is a

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generalized form of fuzzy set. It can describe uncertainty and vagueness case in which the available information is insufficient and incomplete. Therefore it is expected to simulate the human decision making process and any activity requiring human expertise and knowledge which are inevitably imprecise or vagueness. In recent years, IFS has been applied to the multi-attribute decision making model and method (Chen, 2012; Hajiagha et al., 2015), the manufacturing inventory model (Chakrabortty et al., 2013), the multi-objective reliability optimization problem (Garg et al., 2014; Rani et al., 2016; Razmi et al., 2016), the pattern 4

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recognition (Nguyen, 2016), the logistics outsourcing provider selection problem (Wan et al., 2015). In this paper, we apply intuitionistic fuzzy set concept to construct a membership function and a non-membership function and then present a score function which can objectively

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depict decision maker’s satisfactory degree. Furthermore, we propose three optimization models and three interactive intuitionistic fuzzy methods for solving MLPPs from different point of view. Finally, numerical experiments are done to illustrate the effectiveness of the proposed methods. Note that, there was a related paper by Huang et al. (2015) that used the intuitionistic fuzzy set to describe the uncertainty of the decision makers and proposed an

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interactive method for solving MLPPs. It is significantly different from ours. In their paper, the objective functions of the leader and the followers are linear and the goal is to maximize the minimal satisfactory level. In our paper, we propose three interactive intuitionistic fuzzy models which are different from the model developed by Huang et al. (2015) for the

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general MLPPs by considering various decision requirements of experts. Also, a new distance function is developed to measure the merits of a satisfactory solution.

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The rest of this paper is organized as follows. In Section 2, we express a formulation of MLPP which will be studied in this paper. In Section 3, three optimization models and

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three interactive intuitionistic fuzzy methods are proposed. In Section 4, a case study for a cloud computing pricing problem is presented to show the applications of the proposed

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models and methods. In Section 5, numerical experiments are implemented to illustrate the

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efficiency of these models and methods. Finally, we conclude this paper in Section 6.

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2.

Formulation of MLPP In the following, we study a t-level programming problem. The formulation is stated as

follows: f1 (x)

(The top level)

where x2 , · · · , xt solve : min

x2 ,··· ,xt

f2 (x)

(The second level)

.. .

where xt solves : xt

ft (x)

s.t. G(x) ≤ 0,

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(The t-th level)

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min

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min

x1 ,x2 ,··· ,xt

where xi ∈ Rni , i = 1, 2, · · · , t, x = (x1 , x2 , · · · , xt ), G : Rn1 × Rn2 × · · · Rnt → Rm , fi and xi are the objective function and the decision variable of the ith decision maker (DMi),

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respectively. S = {x|G(x) ≤ 0} is the feasible region of problem (1).

Interactive intuitionistic fuzzy methods

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3.

In this section, we first apply the intuitionistic fuzzy set concept to construct a member-

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ship function and a non-membership function, and then present a score function to depict the satisfactory degrees of decision makers. An illustrative example is adopted to illustrate

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the advantages of an intuitionistic fuzzy method in decision making process. Finally, we

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propose three interactive intuitionistic fuzzy methods for MLPPs.

3.1.

Membership function and non-membership function

Definition 3.1. (Atanassov, 1999; Chakrabortty et al., 2013) Let X = {x1 , x2 , · · · , xn } be a finite universal set. Then an Atanassov’s IFS A in X is a set of ordered triples, i.e. A = {< xi , µA (xi ), νA (xi ) >: xi ∈ X},

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where µA (xi ) : X → [0, 1] and νA (xi ) : X → [0, 1] are functions. The values of µA (xi ) and νA (xi ) respectively represent the membership degree and the non-membership degree of xi in X and they meet the condition 0 ≤ µA (xi ) + νA (xi ) ≤ 1. Membership functions express the acceptable degrees of decision makers for a solution.

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A membership function is stated as follows:    1, fi (x) ≤ L0i ,    Ui0 −fi (x) µi (fi (x)) = , L0i ≤ fi (x) ≤ Ui0 , Ui0 −L0i      0, fi (x) ≥ Ui0 ,

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L0i (i = 1, 2, · · · , t) can be derived by

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where the membership degrees of fi (x) at L0i and Ui0 (i = 1, 2, · · · , t) are 1 and 0, respectively.

L0i = min fi (x), i = 1, 2, · · · , t. x∈S

(3)

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If the feasible region S of (1) is bounded, Ui0 (i = 1, 2, · · · , t) can be obtained by solving

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the following optimization problems:

Ui0 = max fi (x). x∈S

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Otherwise, denote the solutions of (3) by x¯i , (i = 1, 2, · · · , t), and assume that Ui0 (i =

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1, 2, · · · , t) are calculated by

Ui0 = max

i∈1,2,··· ,t

fi (¯ xi ), i = 1, 2, · · · , t.

(5)

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Non-membership functions express the rejective degrees of decision makers for a solution.

The rejective degrees of the upper bounds Ui0 , i = 1, 2, · · · , t, which are the worst values for decision makers, are 1. Then a non-membership function of fi (x) is defined as follows:    0, fi (x) ≤ L1i ,    fi (x)−L1i (6) νi (fi (x)) = , L1i ≤ fi (x) ≤ Ui1 , Ui1 −L1i      1, fi (x) ≥ Ui1 , 7

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where Ui1 = Ui0 . Let L1i = L0i + ρi (Ui0 − L0i ), where 0 < ρi < 1, i = 1, 2, · · · , t are determined by decision makers. In a multilevel programming problem, the fuzzy goals of decision makers are to maximize the membership functions and to minimize the non-membership functions. In order to reach

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these fuzzy goals, we present score functions si (fi (x)) = µi (fi (x)) − νi (fi (x)), i = 1, 2, · · · , t to express the satisfactory degrees of decision makers and maximize the score functions.

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An illustrative example

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To illustrate the advantages of an intuitionistic fuzzy method in decision making process, we show a selection problem modified from Lai (1996) in this section and compare an intuitionistic fuzzy method with Stackelberg decision and a fuzzy method for this problem. A government (the leader) chooses a tax rate x from potential rates {a1 , a2 , a3 } to tax a firm (the follower) which selects a product y from {b1 , b2 , b3 } to maximize its objective.

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¯ The second entry of Table 1 shows the ordered profit pairs of the feasible decision set S. the ordered pairs represents the objective value f1 (x, y) of the government and the first

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entry expresses the objective value f2 (x, y) of the firm. It is easy to check that (b1 , a2 ) is the Stackelberg solution with the profit pair (9, 7). Note that the follower dominates the

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Stackelberg solution. Unfortunately, it is inconsistent with the authority of the leader.

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Table 1: A normal form of a two-person, noncooperative game a1

a2

a3

b1

(8, 7)

(9, 7)

(5, 11)

b2

(12, 6)

(7, 8)

(8, 9)

b3

(11, 8)

(6, 10)

(10, 5)

To obtain a satisfactory solution for the leader and the follower, we first use the fuzzy method proposed by Lai (1996) to solve this selection problem. The feasible objective values of the government and the firm are {5, 6, 7, 8, 9, 10, 11} and {5, 6, 7, 8, 9, 10, 11, 12}, respectively. The government is absolutely satisfied with the objective value 11, then the satisfac8

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Table 2: Membership degree, non-membership degree and satisfactory degree of feasible decisions a2

a3

(8, 7)

(9, 7)

(5, 11)

(µF , µL )

(0.43, 0.33)

(0.57, 0.33)

(νF , νL )

(0.25, 0.33)

(0, 0.33)

(sF , sL )

(0.18, 0)

(0.57, 0)

(12, 6)

(7, 8)

b2 (µF , µL )

(1, 0.17)

(νF , νL )

(0, 0.67)

(sF , sL )

(1, -0.5) (11, 8)

(1, 0)

(-1, 1) (8, 9)

(0.29, 0.5)

(0.43, 0.67)

(0.5, 0)

(0.25, 0)

(-0.21, 0.5)

(0.18, 0.67)

(6, 10)

(10, 5)

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b3

(0, 1)

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b1

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a1

(0.86, 0.5)

(0.14, 0.83)

(0.71, 0)

(νF , νL )

(0, 0)

(0.75, 0)

(0, 1)

(sF , sL )

(0.86, 0.5)

(-0.61, 0.83)

(0.71, -1)

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(µF , µL )

Table 3: Result comparisons of different methods Fuzzy method

Intuitionistic fuzzy method

Solution

(9, 7)

(6, 10)

(8, 9)

(µF , µL )

(0.57, 0.33)

(0.14, 0.83)

(0.43, 0.67)

(νF , νL )

(0, 0.33)

(0.75, 0)

(0.25, 0)

(sF , sL )

(0.57, 0)

(-0.61, 0.83)

(0.18, 0.67)

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Stackelberg decision

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tory degree is 1. Conversely, 5 is the minimum profit value which is not satisfied with the government, so its satisfactory degree is 0. The membership degrees µL (f1 (x, y)) (µL for short) of the feasible objective values of the government are {0, 1/6, 2/6, 3/6, 4/6, 5/6, 1}. Similarly, the membership degrees µF of the feasible objective values of the firm are {0, 1/7, 2/7, 3/7,

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4/7, 5/7, 6/7, 1}. The government determines an initial minimal satisfactory level α and would adjust it in the decision making process. The optimal choice of the firm is ¯ {y ∗ |f2 (x∗ , y ∗ ) ≥ f2 (x, y), µL ≥ α, (x, y) ∈ S}.

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Table 2 displays membership degrees of decision makers. When α = 0.9, the feasible decision

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is (b1 , a3 ) and µL = 1. Note that µF = 0. When the leader decreases α to 0.8, the feasible decision is (b3 , a2 ) with µF = 0.14, which is a solution of the fuzzy method. In the following, we propose an intuitionistic fuzzy method to solve this selection problem. In general, the government and the firm are satisfied with large objective values. We assume

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the government’s rejective degrees are 0 when its objective values are larger than 8 and its rejective degree is 1 at the objective value 5; the firm’s rejective degree is 0 when its objective

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values are larger than 9 and its rejective degree is 1 at the objective value 5. Furthermore, we adopt a linear proportional rejective degree to express the objective values between 5

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and 8 of the government and the objective values between 5 and 9 of the firm. Table 2 shows the leader’s non-membership degree νL , the follower’s non-membership degree νF , the

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leader’s satisfactory degree sL = µL − µL and the follower’s satisfactory degree sF = µF − νF . Note that, in Table 2, the negative value of the satisfactory degree means that the acceptable

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degree is less than the rejective degree, so the decision maker is not satisfied with the feasible decision and will reject it. Clearly, a satisfactory solution should satisfy sL ≥ sF ≥ 0.

(8)

This condition is to guarantee the dominant position of the leader and to assure a solution such that the acceptable degree is higher than the rejective degree. It is easy to obtain 10

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that a satisfactory solution solved by the intuitionistic fuzzy method is (b2 , a3 ) with profit pair (8, 9). Finally, in Table 3, we compare the performance of the solutions of Stackelberg decision, fuzzy method and intuitionistic fuzzy method. We also can conclude that: (i) The satisfactory degree of the leader in Stackelberg decision is less than that of the follower. It

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means that the follower’s decision power dominates the leader’s. (ii) The satisfactory degree of the follower in fuzzy method is negative. It represents that the follower is not satisfied with the solution since the rejective degree is larger than acceptable degree. (iii) Neither Stackelberg decision (b1 , a2 ) nor fuzzy solution (b3 , a2 ) meet the condition (8). However, a satisfactory solution satisfying the condition (8) is obtained by intuitionistic fuzzy method.

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Therefore, we conclude that a satisfactory solution obtained by intuitionistic fuzzy method is more rational than that obtained by the other two methods for this selection problem.

3.3.

Three interactive intuitionistic fuzzy methods (IIFMs)

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The goal of each level in the t-level programming problem (1) is to minimize its objective function. Moreover, each level cannot reach its minimization objective regardless of the

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other participants decisions. Therefore, we argue that the fuzzy goal of each level such as the objective function fi (x) (i = 1, 2, · · · , t) should be larger than or equal to a threshold. In

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decision making process, each decision maker specifies a minimal satisfactory level θi ∈ [0, 1] (i = 1, 2, · · · , t) before negotiating with the other decision makers, and si (fi (x)) should be

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more than θi . Due to the hierarchical structure of MLPPs, the upper level dominates the lower levels. Thus the minimal satisfactory level of the upper level should be not less than

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that of the lower level. Now, we propose the following three models. For the first model (Model I), we maximize a minimal satisfactory degree to reach the

fuzzy goals of decision makers. Define η = min{s1 (f1 (x)), s2 (f2 (x)), · · · , st (ft (x))},

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(9)

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and solve the following initial optimization problem max η x,η

x ∈ S, η ∈ [0, 1]

(10)

si (fi (x)) ≥ η, i = 1, 2, · · · , t,

(11)

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s.t.

si−1 (fi (x)) ≥ si (fi (x)), i = 2, · · · , t,

(12)

µi (fi (x)) ≥ νi (fi (x)) ≥ 0, i = 1, 2, · · · , t,

(13)

µi (fi (x)) + νi (fi (x)) ≤ 1, i = 1, 2, · · · , t.

(14)

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Constraint (12) ensures that the upper level dominates the lower levels. Constraint (13) guarantees that the membership degree is not less than the non-membership degree. Note that, the model of Huang et al. (2015) does not touch the constraint (12). They consider the condition (12) in the termination criteria, which will increase the subjectivity and iterations. Therefore, Model I and the model proposed by Huang et al. (2015) are different, though

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some formal resemblance between two models.

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For the second model (Model II), we maximize the sum of satisfactory degrees of decision makers to attain their fuzzy goals. Consider the following optimization problem max

si (fi (x))

i=1

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x

t P

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s.t.

x ∈ S, si−1 (fi (x)) ≥ si (fi (x)), i = 2, · · · , t,

(15)

µi (fi (x)) ≥ νi (fi (x)) ≥ 0, i = 1, 2, · · · , t, µi (fi (x)) + νi (fi (x)) ≤ 1, i = 1, 2, · · · , t.

For the third model (Model III), we denote p t X (fi (x) − Ui0 )2 p γ(x) = , (fi (x) − L0i )2 + (Ui0 − L0i )2 i=1

where L0i is the ideal value of DMi. The closer x gets to L0i , the bigger the value of p p (fi (x) − Ui0 )2 / (fi (x) − L0i )2 + (Ui0 − L0i )2 . Therefore we maximize γ(x) to reach the fuzzy goals of decision makers:

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max γ(x) x

s.t.

x ∈ S, si−1 (fi (x)) ≥ si (fi (x)), i = 2, · · · , t,

(16)

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µi (fi (x)) ≥ νi (fi (x)) ≥ 0, i = 1, 2, · · · , t, µi (fi (x)) + νi (fi (x)) ≤ 1, i = 1, 2, · · · , t.

The above three optimization models achieve fuzzy goals of decision makers from different point of view. Model I maximizes the minimal satisfactory degree in the objective function and ensures the satisfactory balance among decision makers. Model II pursues the

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maximization of overall satisfactory level of decision makers. Model III also considers the overall benefits of decision makers and tries to find a solution x∗ such that fi (x∗ ) is closest to the idea value of DMi. According to the actual needs, decision makers can choose one of the three models to realize their goals.

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In a real decision making process, decision makers determine the minimal satisfactory levels θi ∈ [0, 1], i = 1, 2, · · · , t before negotiations. Thus the satisfactory degrees of the

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above three models should be not less than θi . Furthermore, we consider the relationship and the overall satisfactory balance among the decision makers, which can be revealed by si+1 (fi+1 (x)) si (fi (x))

(i = 1, 2, · · · , t − 1). Here σi ∈ [0, 1]. The acceptable intervals [σi0 , σi1 ] is

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σi =

determined by DMi before the negotiations. Therefore, a satisfactory solution x∗ should

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satisfy the following criteria. Termination Criteria.

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(i) si (fi (x∗ )) ≥ θi , for all i = 1, 2, · · · , t; (ii) σi ∈ [σi0 , σi1 ], for all i = 1, 2, · · · , t − 1. If a solution x∗ does not meet the termination criteria, decision makers will adjust their

satisfactory levels θi , i = 1, 2, · · · , t to obtain another solution. The details of update strategies are stated as follows: Update Strategies. (a) If condition (i) for DMi is not satisfied, DMi should reduce his satisfactory levels θi ; 13

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(b)

If σi < σi0 , DMi should reduce his satisfactory level θi ; If σi > σi1 , DM(i + 1) should

reduce his satisfactory level θi+1 . Suppose that DMj is located at the bottom level of the decision makers who are not 0

satisfied with x∗ . Update his satisfactory level to θj , and then the three models can be

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transformed into the following problems, respectively. Model I: max η x,η

s.t.

x ∈ S, η ∈ [0, 1],

si (fi (x)) ≥ η, i = 1, 2, j − 1, j + 1, · · · , t, 0

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sj (fj (x)) ≥ θj ,

(17)

si−1 (fi (x)) ≥ si (fi (x)), i = 2, · · · , t,

µi (fi (x)) ≥ νi (fi (x)) ≥ 0, i = 1, 2, · · · , t, Model II: max x

si (fi (x))

i=1

x ∈ S,

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s.t.

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µi (fi (x)) + νi (fi (x)) ≤ 1, i = 1, 2, · · · , t.

0

sj (fj (x)) ≥ θj ,

(18)

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si−1 (fi (x)) ≥ si (fi (x)), i = 2, · · · , t,

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µi (fi (x)) ≥ νi (fi (x)) ≥ 0, i = 1, 2, · · · , t,

µi (fi (x)) + νi (fi (x)) ≤ 1, i = 1, 2, · · · , t.

Model III:

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max γ(x) x

s.t.

x ∈ S, 0

sj (fj (x)) ≥ θj ,

(19)

si−1 (fi (x)) ≥ si (fi (x)), i = 2, · · · , t, µi (fi (x)) ≥ νi (fi (x)) ≥ 0, i = 1, 2, · · · , t, µi (fi (x)) + νi (fi (x)) ≤ 1, i = 1, 2, · · · , t.

When decision makers are satisfied with a solution x∗ , the decision making process is

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terminated. Otherwise, decision makers should adjust their satisfactory level according to the update strategies until attaining a satisfactory solution. Based on Model I, Model II and Model III, we respectively present three interactive intuitionistic fuzzy methods as follows.

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Method I Initialization: Decision makers set the initial minimal satisfactory levels θi , i = 1, 2, · · · , t and the acceptable interval [σi0 , σi1 ] of σi , (i = 1, 2, · · · , t − 1).

Step 1. DMi determines L0i , Ui0 , ρi and calculates his membership function µi (fi (x)), non-membership function νi (fi (x)) and score function si (fi (x)) = µi (fi (x)) − νi (fi (x)),

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i = 1, 2, · · · , t.

Step 2. Solve the initial optimization problem and obtain a solution (x∗ , η ∗ ). Step 3. If (x∗ , η ∗ ) meets the termination criteria, decision makers obtain a satisfactory solution and the algorithm stops.

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Step 4. If DMj is located at the bottom level of the decision makers who are not satisfied 0

with (x∗ , η ∗ ). Update his satisfactory level to θj according to the update strategies and go to

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Step 5.

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Step 5. Solve problem (17) to get a new solution (x∗ , η ∗ ). Return to Step 3. Method II Initialization: Decision makers set the initial minimal satisfactory levels θi , i = 1, 2, · · · , t

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and the acceptable interval [σi0 , σi1 ] of σi , (i = 1, 2, · · · , t − 1). Step 1. DMi determines L0i , Ui0 , ρi and calculates his membership function µi (fi (x)),

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non-membership function νi (fi (x)) and score function si (fi (x)) = µi (fi (x)) − νi (fi (x)), i = 1, 2, · · · , t. Step 2. Solve problem (15) and get a solution x∗ . Step 3. If x∗ satisfies the termination criteria, decision makers attain a satisfactory solution and the algorithm stops. Step 4. If DMj is located at the bottom level of the decision makers who are not satisfied

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0

with x∗ . Update his satisfactory level to θj according to the update strategies and go to Step 5. Step 5. Solve problem (18) to get a new solution x∗ . Return to Step 3.

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Method III Initialization: Decision makers set the initial minimal satisfactory levels θi , i = 1, 2, · · · , t and the acceptable interval [σi0 , σi1 ] of σi , (i = 1, 2, · · · , t − 1).

Step 1. DMi determines L0i , Ui0 , ρi and calculates his membership function µi (fi (x)), non-membership function νi (fi (x)) and score function si (fi (x)) = µi (fi (x)) − νi (fi (x)),

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i = 1, 2, · · · , t.

Step 2. Solve problem (16) and obtain a solution x∗ .

Step 3. If x∗ satisfies the termination criteria, decision makers attain a satisfactory solution and the algorithm stops.

Step 4. If DMj is located at the bottom level of the decision makers who are not satisfied 0

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with x∗ . Update his satisfactory level to θj according to the update strategies and go to Step 5.

A case study: cloud computing pricing problem

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4.

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Step 5. Solve problem (19) to get a new solution x∗ . Return to Step 3.

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In this section, a cloud computing pricing problem is modeled as a trilevel programming problem to illustrate the applications of the proposed solution methods (IIFMs).

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In the cloud computing market, software as a service (SaaS) offers cloud softwares resources for end-users (individual, enterprises or government) and uses infrastructure as a service (IaaS) from IaaS providers. End-users choose to use cloud resources or their internal resources on the basis of their prices. For example, Drew Houston and Arash Ferdowsi cofounded Dropbox in 2007. Dropbox offers users and enterprises storage software resources and uses IaaS from Amazon S3 without management and maintenance of basic resources to lower a lot of cost. In the following, we consider the pricing problems of cloud storage

16

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software resources (SaaS) such as Dropbox, box.com, OneDrive, and SugarSync, and basic storage resources (IaaS) such as Amazon simple storage service (Amazon S3). The aims of the IaaS provider and the SaaS provider are respectively to maximize their profits. The end-users want to purchase economic and efficient cloud storage software resources. These

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decision makers interact together. End-users react to the pricing of the SaaS which is set by the price of the IaaS and is affected by the end-users purchases; the pricing of the IaaS provider is also affected by the users purchases. To address these relationship and model the pricing problems, we propose a trilevel programming model. As the IaaS provider offers the basic storage capacity and the SaaS provider offers end-users with software resources, the

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IaaS is the top level (DM1), the SaaS is the middle level (DM2), and the end-users are the bottom level (DM3). The trilevel pricing model is built as follows. max

f1 (pI , pS , x) =

pI ,pS ,x

i∈N

pI − 0.01 ≥ 0,

where pS and x solve:

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max f2 (pI , pS , x) = s.t.

pI xi1 −

M

s.t.

pS ,x

P

pI ≤ 0.88pS ,

P

i∈N

pS xi1 −

P

i∈N

P

0.01xi1

(20)

i∈N

0.05xi1 −

P

pI xi1

i∈N

(21)

CE

where x solves:

PT

pS ≤ 0.635,

min f3 (pI , pS , x) = x

s.t.

P

pS xi1 +

i∈N

10 ≤ xi ≤ ηi ,

∀i ∈ N.

P

i∈N

0.635xi2

(22)

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The goals of the IaaS provider and SaaS provider are respectively to maximize their

profits f1 and f2 . End-users seek the cost minimization. pI and pS are the prices of the IaaS

and the SaaS, respectively. If an end-user is not satisfied with the price of SaaS, he will use traditional tools for storing instead of SaaS resources. Let xi1 be the demands of end-users for SaaS, xi2 be the service capacity for using internal resources and xi = xi1 + xi2 be the total demand amount of user i for the application service. N is the number of end-users, x = (x11 , x21 , · · · , xN 1 , x12 , x22 , · · · , xN 2 ). pI , pS , x are respectively the decision variables of 17

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the top level, middle level and bottom level. The parameters of the participants are set based on box.com (2016) and Amazon (2016). We set ηi = 105 T, and |N | = 50. Using the proposed IIFMs to solve this pricing problem, we get the following results. For Method I, the satisfactory degrees of each decision maker are s1 (f1 (pI , pS , x)) =

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0.3411, s2 (f2 (pI , pS , x)) = 0.32 and s3 (f3 (pI , pS , x)) = 0.32. The pricing of the IaaS provider and the SaaS provider are pI = $0.1942 and pS = $0.4250, respectively. And the corresponding function values of cloud providers are f1 = $9.2024 × 105 , f2 = $9.0326 × 105 and P PN f3 = $2.1234 × 106 . N i=1 xi1 / i=1 xi = 1.0000. For Method II, the satisfactory degrees of each decision maker are s1 (f1 (pI , pS , x)) =

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0.3527, s2 (f2 (pI , pS , x)) = 0.32 and s3 (f3 (pI , pS , x)) = 0.32. The pricing of the IaaS provider and the SaaS provider are pI = $0.2022 and pS = $0.4330, respectively. And the corresponding function values of cloud providers are f1 = $9.4116 × 105 , f2 = $8.8527 × 105 and PN P f3 = $2.1241 × 106 . N i=1 xi = 0.9987. i=1 xi1 /

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For Method III, the satisfactory degrees of each decision maker are s1 (f1 (pI , pS , x)) =

0.3411, s2 (f2 (pI , pS , x)) = 0.32 and s3 (f3 (pI , pS , x)) = 0.32. The pricing of the IaaS provider

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and the SaaS provider are pI = $0.2029 and pS = $0.4337, respectively. And the corre-

PT

sponding function values of cloud providers are f1 = $9.2029 × 105 , f2 = $9.0329 × 105 and PN P f3 = $2.1234 × 106 . N i=1 xi = 1.0000. i=1 xi1 /

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The results show that the three methods can solve the cloud computing pricing problems. P PN Comparing the values of f1 , f2 , f3 and N i=1 xi1 / i=1 xi , we find that Method I can obtain more profit for SaaS provider at a lower price. So Method I is more appropriate than other

AC

methods for cloud services pricing. Furthermore, to present the real situations, let ηi be randomly generated in the ranges

[0, 105 ] T. Using the three methods, we can obtain the following results. For Method I, the satisfactory degrees of each decision maker are s1 (f1 (pI , pS , x)) = 0.3409, s2 (f2 (pI , pS , x)) = 0.32 and s3 (f3 (pI , pS , x)) = 0.32. The pricing of the IaaS provider and the SaaS provider are pI = $0.1942 and pS = $0.4250, respectively. And the corresponding function values of cloud providers are f1 = $5.1707 × 105 , f2 = $5.0740 × 105 and 18

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f3 = $1.1937 × 106 . For Method II, the satisfactory degrees of each decision maker are s1 (f1 (pI , pS , x)) = 0.3529, s2 (f2 (pI , pS , x)) = 0.32 and s3 (f3 (pI , pS , x)) = 0.32. The pricing of the IaaS provider and the SaaS provider are pI = $0.2058 and pS = $0.4366, respectively. And the corre-

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sponding function values of cloud providers are f1 = $4.8101 × 105 , f2 = $4.4410 × 105 and f3 = $1.0729 × 106 .

For Method III, the satisfactory degrees of each decision maker are s1 (f1 (pI , pS , x)) = 0.3497, s2 (f2 (pI , pS , x)) = 0.32 and s3 (f3 (pI , pS , x)) = 0.32. The pricing of the IaaS provider and the SaaS provider are pI = $0.2029 and pS = $0.4337, respectively. And the corre-

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sponding function values of cloud providers are f1 = $4.3409 × 105 , f2 = $4.0680 × 105 and f3 = $9.7686 × 105 .

Comparing the above three results, we find that Method I has the lower prices for cloud services and bring more profit for IaaS and SaaS. These mean that the pricing of Method I

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can attract more users to use cloud services. As is well-known, cloud providers will attract more users to use cloud services to obtain more profit. So Method I is more appropriate for

Numerical experiments

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5.

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cloud computing pricing problems than other methods.

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To illustrate the efficiency of the proposed interactive intuitionistic fuzzy methods, we

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conduct the following numerical experiments. Note that, some authors use a distance funcP 1 tion { ti=0 [1 − µi (fi (x))]2 } 2 to select a priority solution (Pramanik and Roy, 2007; Zheng P 1 et al., 2014). Similarly, we develop a new distance function D(x) = { ti=0 [1 − si (fi (x))]2 } 2

to select a priority solution. The smaller the value of the distance function is, the better a solution will be. We respectively denote µi (fi (x∗ )), νi (fi (x∗ )) and si (fi (x∗ )) by µi , νi and si

for short.

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Example 1 (Zheng et al., 2014) max F1 (x, y, z) = −18x1 + 10x2 + 11y1 − 11y2 + 23z1 + 40z2 , x

max F2 (x, y, z) = −35x1 − 9x2 + 20y1 − 44y2 + 10z1 + 7z2 , y,z

s.t.

47x1 − 14x2 − y1 + 4y2 + z1 − 49z2 ≤ 1.5,

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−23x1 + 2x2 + 45y1 − 35y2 + 12z1 + 41z2 ≤ 13.5, −9x1 − 18x2 + 12y1 + 13y2 + 37z1 − 11z2 ≤ 5.5, 6x1 − 19x2 − y1 − 2y2 − 49z1 − 11z2 ≤ −43.5,

−31x1 − 8x2 + 2y1 + 17y2 + 47z1 − 25z2 ≤ 6.3,

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46x1 + 3x2 − 28y1 + 17y2 − 36z1 − 3z2 ≤ 22.5,

−45x1 + 34x2 − 44y1 + 44y2 + 16z1 − 2z2 ≤ 17, 29x1 − 13x2 + 38y1 + 19y2 − 2z1 + 7z2 ≤ 39,

13x1 + 10x2 + 27y1 − 29y2 − 49z1 − 38z2 ≤ −38, xi ≥ 0, yi ≥ 0, zi ≥ 0, i = 1, 2.

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This is a maximization type problem, and is transformed into a minimization problem. It is easy to obtain that L01 = −51.3107, L02 = 13.5330, U10 = U11 = −22.0674, U20 = U21 =

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56.2629. Set ρ1 = ρ2 = 0.3, then L11 = −48.3864 and L12 = 17.8060. Using Model I, we get a solution: (x∗ , y ∗ , z ∗ ) = (0.8937, 1.1266, 0.0000, 0.0735, 1.0465, 0.5321), µ1 = 0.6038,

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µ2 = 0.6038, s1 = s2 = 0.4664, D = 0.7547, F1 = 39.7243, F2 = −30.4630. For Method II, solving problem (15), we obtain a solution: (x∗ , y ∗ , z ∗ ) = (0.8937, 1.1266,

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0.0000, 0.0735, 1.0465, 0.5321), µ1 = 0.6038, µ2 = 0.6038, s1 = s2 = 0.4664, D = 0.7547, F1 = 39.7243, F2 = −30.4630.

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For Method III, solving problem (16), we get a solution: (x∗ , y ∗ , z ∗ ) = (0.8937, 1.1266,

0.0000, 0.0735, 1.0465, 0.5321), µ1 = 0.6038, µ2 = 0.6038, s1 = s2 = 0.4664, D = 0.7547, F1 = 39.7243, F2 = −30.4630. We compare the proposed three methods with the methods in Wan et al. (2008) and Zheng et al. (2014) for solving Example 1, and the numerical results are listed in Table 4 and Table 5. For considering the satisfactory balance between the two levels, we assume σ1 should be in the acceptable interval [0.8, 1]. 20

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Table 4: Comparisons of the solutions of Example 1 based on different methods Method

x∗

y∗

z∗

Method I

(0.8937,1.1266)

(0.0000,0.0735)

(1.0465,0.5321)

39.7243

-30.4630

Method II

(0.8937,1.1266)

(0.0000,0.0735) (1.0465,0.5321) 39.7243

-30.4630

Method III

(0.8937,1.1266)

(0.0000,0.0735)

(1.0465,0.5321)

39.7243

-30.4630

(Zheng et al., 2014)

(0.8961,1.1275)

(0.0000,0.0749)

(1.0477,0.5343)

39.7888

-30.5918

(Wan et al., 2008)

(0.8802,1.0951) (0.0000,0.0922) (1.0260,0.5481) 39.6134

-30.6250

(Huang et al., 2015)

(0.8903,1.1253)

-30.2807

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F1

(1.0448,0.5290)

39.6329

ED

M

(0,0.0714)

F2

PT

Table 5: Comparisons of satisfactory degrees of Example 1 based on different methods s1

s2

D

σ1

Method I

0.4664

0.4664

0.7547

1

Method II

0.4664

0.4664

0.7547

1

Method III

0.4664

0.4664

0.7547

1

(Zheng et al., 2014)

0.4718

0.4592

0.7560

0.9733

(Wan et al., 2008)

0.4573

0.4573

0.7675

1

(Huang et al., 2015)

0.4588

0.4767

0.7528

1.0390

AC

CE

Method

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Table 4 shows the comparisons of the satisfactory solutions. Table 5 displays the comparisons of decision makers’ satisfactory degrees s1 , s2 and the value of distance function D, the ratio of the values of score functions σ1 . From Table 5, it is easy to find that σ1 in Huang et al. (2015) is larger than 1. This expresses that the satisfactory degree of the lower

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level is higher than that of the upper level and the lower level dominates the upper level. In addition, the values of D in our proposed methods are smaller than that in Wan et al. (2008) and Zheng et al. (2014). So our proposed methods are better than the methods in Wan et al. (2008), Zheng et al. (2014) and Huang et al. (2015).

Next, we further present five numerical examples to illustrate the efficiency of our pro-

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posed methods. To obtain a satisfactory solution, assume that the ratio of the values of score functions σi , i = 1, 2, · · · , t − 1 should be in the acceptable interval [0.8, 1]. If σi is larger than 1, it represents that DMi + 1 dominates DMi, which is in contradiction with the hierarchical structure.

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Example 2 (Pramanik and Roy, 2007)

max Z1 = 7x1 + 3x2 − 4x3 + 2x4 ,

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x1 ,x2

max Z2 = x2 + 3x3 + 4x4 , x3

max Z3 = 2x1 + x2 + x3 + x4 ,

PT

x4

AC

CE

s.t.

x1 + x2 + x3 + x4 ≤ 5, x1 + x2 − x3 − x4 ≤ 2, x1 + x2 + x3 ≥ 1, −x1 + x2 + x3 ≤ 1, x1 − x2 + x3 + 2x4 ≤ 4, x1 + 2x3 + 3x4 ≤ 3, x4 ≤ 2, x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0.

Table 6 shows the comparisons of the satisfactory solutions of Example 2. Table 7 displays the comparisons of satisfactory degrees of Example 2. σ1 and σ2 in our proposed methods 22

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Table 6: Comparisons of satisfactory solutions of Example 2 based on different methods x∗

y∗

Z1

Z2

Z3

Method I

(1.0490,1.6079)

(0.0199,0.6371)

13.3617

4.2158

4.3629

Method II

(1.0327,1.6230)

(0,0.6558)

13.4097

4.2461

4.3442

Method III

(1.0327,1.6230)

(0,0.6558)

13.4097

4.2461

4.3442

(Pramanik and Roy, 2007)(I)

(1.106,1.525)

(0,0.631)

13.58

4.05

4.37

(Pramanik and Roy, 2007)(IIa)

(0.857,1.857)

(0,0.714)

13

4.71

4.28

(Pramanik and Roy, 2007)(IIb)

(0.857,1.857)

(0,0.714)

13

4.71

4.28

(Sinha, 2003a,b)

(1.59,1.08)

(0.62,0.06)

12.01

3.18

4.94

(Shih et al., 1996)

(0.857,1.857)

(0,0.714)

11.7

3.02

4.94

ED

M

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Method

Table 7: Comparisons of satisfactory degrees of Example 2 based on different methods s1

s2

s3

D

σ1

σ2

0.8100

0.7800

0.7749

0.3677

0.9630

0.9935

0.8150

0.7928

0.7650

0.3638

0.9728

0.9649

Method III

0.8150

0.7928

0.7650

0.3638

0.9728

0.9649

(Pramanik and Roy, 2007)(I)

0.8327

0.7096

0.7776

0.4023

0.8522

1.0958

(Pramanik and Roy, 2007)(IIa)

0.7721

0.9426

0.7337

0.3551

1.2208

0.7784

(Pramanik and Roy, 2007)(IIb)

0.7721

0.9426

0.7337

0.3551

1.2208 0.7784

(Sinha, 2003a,b)

0.6691

0.3427

0.9850

0.7361

2.8742

(Shih et al., 1996)

0.6368

0.2751

0.9850

0.8109

3.5805 0.8232

Method I

AC

CE

Method II

PT

Method

23

0.7473

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are all in the acceptable interval [0.8, 1]. σ1 in Pramanik and Roy (2007)(IIa), Pramanik and Roy (2007) (IIb), Shih et al. (1996), Sinha (2003a) and Sinha (2003b) are larger than 1. It means that the satisfactory degree of the second level is higher than that of the first level, i.e. the second level dominates the first level. σ2 in Pramanik and Roy (2007)(I) is larger

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than 1. It expresses that the third level has a high satisfactory degree than the second level, i.e. the third level dominates the second level. These results are at variance with a fact in MLPPs that the upper level has more priority than the lower level.

In addition, comparing our proposed three methods, the value of D of Method II and Method III is smaller than that of Method I. Thus, a satisfactory solution of Method II and

Example 3 (Zheng et al., 2014)

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Method III is better than that of Method I.

max F1 (x, y) = 5x1 + 2x2 + 4x3 x

where for a given x = (x1 , x2 )T , y = (x3 , x4 )T solves,

M

max F2 (x, y) = 3x2 + 5x3 − 2x4 y

s.t. 2x1 + 2x2 + 2x3 + 4x4 ≤ 8,

ED

x1 + x2 + x3 ≤ 2, x2 + x3 + x4 ≤ 3,

PT

x1 ≤ 4, x2 ≤ 4,

x3 ≤ 2, x4 ≤ 2

CE

x1 , x2 , x3 , x4 ≥ 0.

Table 8 presents the comparisons of the satisfactory solutions of Example 3. Table 9

AC

exhibits the comparisons of satisfactory degrees of Example 3. σ1 in our proposed methods are all in the acceptable interval [0.8, 1]. σ1 in Zheng et al. (2014) is larger than 1. This

states that the satisfactory degree of the second level is higher than that of the first level, i.e. the second level dominates the first level. This result is at variance with a fact in MLPPs that the upper level has more priority than the lower level. σ1 in Arora and Gupta (2009) is lower than 0.8, which expresses that the ratio of the second level and the first level is lower than the acceptable value. 24

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Table 8: Comparisons of satisfactory solutions of Example 3 based on different methods Method

x∗

y∗

Method I

(0.4375,0)

Method II

F2

(1.5625,0.0000)

8.4375

7.8125

(0.4375,0)

(1.5625,0.0000)

8.4375

7.8125

Method III

(0.4375,0)

(1.5625,0)

8.4375

7.8125

(Zheng et al., 2014)

(0,0)

(2,0)

8

10

(Arora and Gupta, 2009)

(1,0)

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F1

9

5

PT

ED

M

(1,0)

Table 9: Comparisons of satisfactory degrees of Example 3 based on different methods s1

s2

D

σ1

Method I

0.7812

0.7812

0.3094

1

Method II

0.7812

0.7812

0.3094

1

Method III

0.7812

0.7812

0.3094

1

(Zheng et al., 2014)

0.8000

1

0.2000

1.2500

(Arora and Gupta, 2009)

0.9000

0.6429

0.3709

0.7143

AC

CE

Method

25

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-34

-7

-44

-22

-10

c2

-13

-25

-1

-5

-27

d1

22

36

4

31

49

d2

1

48

33

37

4

c3

-11

-33

-31

-14

-7

c4

13

25

1

5

27

d3

38

5

42

10

8

d4

1

48

33

37

4

A1

-6

38

-30

24

-24

A2

-10

-41

42

20

2

-29

-22

16

43

-15

15

3

-9

11

24

22

-28

-22

43

-45

-37

30

-45

-18

-20

-27

-101

5

44

46

11

-49

-5

-11

49

6

16

67

36

47

-39

20

-23

-45

34

-43

10

-11

-8

21

40

-17

36

50

-4

11

-28

36

16

96

32

18

-7

-15

-23

-41

-44

-31

42

17

-31

3

-41

-14

15

-8

-39

35

-15

-20

22

-37

-31

-36

27

-17

17

-8

22

44

-4

45

35

3

-6

5

-23

-22

-31

-42

31

4

-21

-61

13

-43

-10

49

-2

34

-13

0

27

-19

21

ED

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c1

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Table 10: Coefficients in Example 4

b 8

AC

CE

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Example 4 (Sakawa and Nishizaki, 2009) min z1 (x) =

c1 x1 +c2 x2 +1 d1 x1 +d2 x2 +1

min z2 (x) =

c3 x1 +c4 x2 +1 d3 x1 +d4 x2 +1

x1

x2

(23)

subject to A1 x1 + A2 x2 ≤ b xij ≥ 0, i = 1, 2, j = 1, · · · , 5,

where x1 = (x11 , · · · , x15 )T , x2 = (x21 , · · · , x25 )T . The coefficients are shown in Table 10. The comparisons of the satisfactory solutions of Example 4 are displayed in Table 11.

Table 12 shows the comparisons of satisfactory degrees of Example 4. σ1 in Sakawa and Nishizaki (2009) is lower than 0.8, which states that the ratio of the lower level and the upper level is lower than the acceptable value. However, σ1 in our proposed methods are all in the acceptable interval [0.8, 1]. Moreover, the satisfactory solutions obtained by our 26

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Table 11: Comparisons of satisfactory solutions of Example 4 based on different methods x1

x2

Method I

(2.7836,0.2411,1.4097,0,1.1008)

(1.0590,0.8272,0.7012,0,0.7248)

Method II

(2.6035,0.3228,1.1359,0,0.8874) (1.3160,0.8296,0.6132,0,1.0416)

Method III

(2.6035,0.3228,1.1359,0,0.8874)

(1.3160,0.8296,0.6132,0,1.0416)

(Sakawa and Nishizaki, 2009)

(2.604,0.323,1.136,0,0.887)

(1.316,0.830,0.613,0,1.041)

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Method

Table 12: Comparisons of satisfactory degrees of Example 4 based on different methods z1

z2

s1

s2

D

σ1

Method I

-1.1307

-0.1411

0.5828

0.5828

0.5901

1

Method II

-1.1307

-0.1411

0.5828

0.5828

0.5901

1

Method III

-1.1307

-0.1411

0.5828

0.5828

0.5901

1

(Sakawa and Nishizaki, 2009)

-1.173

-0.059

0.7000

0.3883

0.6813

0.5547

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Method

M

proposed three methods have the same satisfactory degree.

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Example 5 (Lachhwani and Poonia, 2012) max Z1 = x1 ,x2

7x1 +3x2 −4x3 +2x4 , x1 +x2 +x3 +1

max Z2 =

x2 +3x3 +4x4 , x1 +x2 +x3 +2

PT

x3

max Z3 =

AC

CE

x4

s.t.

2x1 +x2 +x3 +x4 , x1 +x2 +x3 +3

x1 + x2 + x3 + x4 ≤ 5, x1 + x2 − x3 − x4 ≤ 2, x1 + x2 + x3 ≥ 1, x1 − x2 + x3 + 2x4 ≤ 4, x1 + 2x3 + 3x4 ≤ 3, x4 ≤ 2, x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0.

Table 13 shows the comparisons of the satisfactory solutions of Example 5. Table 14 exhibits the comparisons of satisfactory degrees of Example 5. s1 and σ1 in Lachhwani and 27

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Table 13: Comparisons of satisfactory solutions of Example 5 based on different methods (x1 , x2 , x3 , x4 )

(Z1 , Z2 , Z3 )

Method I

(0.9425,0.7504,0,1.0287)

(4.0500,1.3175,0.7808)

Method II

(0.9425,0.7504,0,1.0287)

(4.0500,1.3175,0.7808)

Method III

(0.9401,0.7401,0,1.0299)

(4.0523,1.3205,0.7799)

(Lachhwani and Poonia, 2012)I

(0.4471,1.69105,0,1.2764)

(3.42738,1.642437,0.7515643)

(Lachhwani and Poonia, 2012)IIb

(1.0000,0,0,1.0000)

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Method

PT

ED

M

(4.5,1.3333,0.75)

Table 14: Comparisons of satisfactory degrees of Example 5 based on different methods s1

s2

s3

D

σ1

σ2

Method I

0.5000

0.4955

0.4955

0.8712

0.9910

1

Method II

0.5000

0.4955

0.4955

0.8712

0.9910

1

Method III

0.5011

0.5000

0.4874

0.8727

0.9978

0.9748

(Lachhwani and Poonia, 2012)I

-0.3894

0.6589

0.2150

1.6319

-1.6921

0.3263

0.5063

0.2000

0.9825

0.7088

0.3950

AC

CE

Method

(Lachhwani and Poonia, 2012)IIb 0.7143

28

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Poonia (2012)I are negative. These indicate that the acceptable degree of the top level is lower than the rejective degree of the top level. The top level is hardly satisfied with this solution for the low satisfactory degree. σ1 and σ2 in Lachhwani and Poonia (2012)IIb are lower than 0.8, which states that the ratios of the lower level and the upper level are not in

the acceptable interval [0.8, 1].

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the acceptable value. However, the values of σ1 and σ2 in our proposed methods are all in

Furthermore, comparing our proposed three methods, the values of D of Method I and Method II are smaller than that of Method III. Thus, a satisfactory solution of Method I

Example 6 (Sakawa et al., 1998)

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and Method II is better than that of Method III.

min z1 (x) = c1 x1 + c2 x2 + c3 x3

x1 ,x2 ,x3

min z2 (x) = c4 x1 + c5 x2 + c6 x3

x2 ,x3

min z3 (x) = c7 x1 + c8 x2 + c9 x3

(24)

M

x3

subject to A1 x1 + A2 x2 + A3 x3 ≤ b

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xj ≥ 0, j = 1, 2, j = 1, · · · , 15,

where x = (x1 , x2 , x3 ). The coefficients are shown in Table 15.

PT

Table 16 displays the comparisons of the satisfactory solutions of Example 6. Table 17 shows the comparisons of satisfactory degrees of Example 5. σ1 and σ2 of our proposed

CE

methods are all in the acceptable interval [0.8, 1]. σ2 in Huang et al. (2015) and Sakawa et al. (1998) are larger than 1, which states that the satisfactory degree of the third level is

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higher than that of the second level, i.e. the third level dominates the second level. This result is at variance with a fact in MLPPs that the upper level has more priority than the lower level.

Comparing our proposed three methods, the value of D of Method I is smaller than that of Method II and Method III. Thus, a satisfactory solution of Method I is better than that of Method II and Method III. The results of the above six examples show that our proposed three methods are efficient 29

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-45 -5 -45 -14 -9

c2

-26 -47 -19 -27 -27

c3

c4

-50 -10 -30 -19 -2

c5

-30 -30 -9 -10 -38

c6

c7

-45 -17 -28 -49 -24

c8

-4 -5 -49 -38 -32

c9

A1

46 -29 -48 31 21

A2

-47 -37 37 32 19

A3

-33 -7 -28 -21 -12 -17 -13 -48 -25 -2

b

-2 -5 -33 -4 -9

21 2 -11 -35 -2

0

-25 6 4 2 5

26

-42 -26 42 31 -1

38 -27 5 -31 14

-38 -29 -5 -47 49

-45 -45 5 10 -40

-111

-26 16 44 6 19

17 27 32 17 -6

-27 1 18 -6 15

88

-48 13 2 -33 19

22 -35 27 -35 -35

-26 -16 37 47 -2

-37

9 -6 12 -17 -32

-8 24 -24 45 -31

16 -9 -19 17 44

12

-9 2 -16 8 32

-6 -25 -25 -8 4

23 41 30 36 -11

45

24 30 42 -26 16

19 -18 -18 9 -34

-46 30 3 -1 -45

-8

-26 -8 0 41 -42

-19 13 -42 49 -27

4 -2 -12 24 -33

-47

-29 16 -16 -4 18

45 -8 21 6 47

43 46 26 22 -5

136

-7 1 -3 38 18

-43 -15 31 -34 23

-35 -34 20 -15 -26

-48

-12 -4 47 0 -4

-18 -19 28 47 -36

-45 20 40 3 -15

19

-12 -46 11 -47 -47

19 30 50 12 -24

13 20 -43 -8 20

-31

5 -2 37 38 0

12 -34 34 28 -40

-18 33 39 14 2

88

49 41 3 12 -48

15 12 32 31 -28

-25 -23 -6 -25 -15

14

-17 -6 34 21 11

5 -28 -46 -15 9

12 49 4 -17 -47

-18

AC

CE

PT

ED

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0 39 12 -14 29

M

c1

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Table 15: Coefficients in Example 6

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Table 16: Comparisons of satisfactory solutions of Example 6 based on different methods Method

(x1 , x2 , x3 )T

z1

z2

z3

Method I

(2.6532,0,0.6444,2.2309,2.3312,

-526.0640

-459.1025

-368.8725

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1.6450,3.7100,0,0,1.0140, 1.59200,0,0,1.3526,0) Method II

(2.7226,0,0,2.2236,2.5651,

-530.6806

-466.0899

-364.1470

-530.6806

-466.0899

-364.1470

-521.7730

-454.6170

-371.2541

(2.5780,0,0.9684,2.2894,2.1415, -515.6722

-451.4191

-371.4737

1.6420,4.4172,0,0,1.0515, 1.2897,0,0,1.5568,0) (2.7226,0,0,2.2236,2.5651,

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Method III

1.6420,4.4172,0,0,1.0515, 1.2897,0,0,1.5568,0) (Huang et al., 2015)

(2.6089,0,0.9619,2.2481,2.1979,

M

1.6543,3.3247,0,0,1.00,

(Sakawa et al., 1998)

ED

1.7204,0,0,1.2583,0.0255)

1.6780,3.2067,0,0,1.0215

CE

PT

1.6617,0,0,1.2751,0.1021)

Table 17: Comparisons of satisfactory degrees of Example 6 based on different methods s1

s2

s3

D

σ1

σ2

Method I

0.9900

0.9825

0.9825

0.0267

0.9924

1

Method II

1

1

0.9678

0.0322

1

0.9678

Method III

1

1

0.9678

0.0322

1

0.9678

(Huang et al., 2015)

0.9807

0.9713

0.9899

0.0361

0.9904

1.0191

(Sakawa et al., 1998)

0.9674

0.9632

0.9906

0.0500

0.9957

1.0284

AC

Method

31

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for solving MLPPs. The proposed three methods characterize various requirements of expert decision makers. Then, they can obtain not only the satisfactory solution and satisfactory balance among decision makers but also the integral benefits of expert decision makers. In addition, the distance function D(x) can be applied to measure the merits of a satisfactory

6.

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solution.

Conclusions

Three interactive intuitionistic fuzzy methods are proposed to attain satisfactory solu-

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tions for MLPPs from different point of view. These methods overcome the subjectivity of interactive fuzzy methods and hence objectively make suggestions in the decision making processes. A score function is developed to depict the satisfactory degree which consists of the membership degree and non-membership degree. In view of numerical experiments, our

M

proposed methods are efficient.

The main contributions of this paper are outlined as follows: (1) Traditional methods

ED

(Bard, 1984; Han et al., 2015; Hosseini and Kamalabadi, 2014; Kolstad and Lasdon, 1990; Lu et al., 2007; White, 1997) emphasize the non-cooperative relationship among decision

PT

makers, but they ignore a solution whether is desired. Fuzzy methods (Arora and Gupta, 2009; Dempe, 2011; Lai, 1996; Pramanik and Roy, 2007; Sakawa et al., 1998; Shih et al.,

CE

1996; Wang et al., 2009; Zheng et al., 2014) take a membership degree as a satisfactory degree which has no objective procedure to determine a membership degree and is subjective

AC

to determine a satisfactory degree. However, our proposed methods use a score function to depict satisfactory degree which is more objective than the above methods for making decisions in expert and intelligent systems. (2) Traditional methods rarely consider the intentions of decision makers. On the other hand, most fuzzy methods and Huang et al. (2015) just consider the maximization of the minimal satisfactory level and satisfactory balance among decision makers, which may be not the main concern of decision makers or cannot obtain a better satisfactory solution. Fortunately, we propose three optimization 32

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models and three interactive intuitionistic fuzzy methods to characterize various intentions of expert decision makers. (3) A criteria depicted by a new distance function is crucial tool to choose the most satisfactory solution for decision makers. The distance functions is constructed by using the idea of intuitionistic fuzzy set and can promote experts to make

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proper decision. Furthermore, our proposed methods can be applied in expert and intelligent systems e.g. supply chain management, production selection and service pricing. For future researches, we will discuss: (1) The variables and constraints considered in this paper are crisp. To be more exact, they may be in other forms such as fuzzy number and intuitionistic fuzzy

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number in practical applications. Therefore our work could be extended to other forms of data in further researches. (2) For simplicity, we only use linear membership function and non-membership function to define the score function. Some nonlinear membership function and non-membership function, such as exponential, hyperbolic, hyperbolic inverse

M

and piecewise linear membership functions, will be considered for meeting decision makers’ special requirements. (3) Some novel skills will be adopted in our methods for addressing

ED

large-scale decision-making problems in expert and intelligent systems. (4) We has provided a case study related to cloud computing pricing problem. Some future researches of modeling

PT

problems in other fields, such as supply management, manufacturing systems and traffic

CE

optimization, could be discussed.

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Acknowledgements This work was supported by the National Nature Science Foundation of China (Nos. 71471140, 11501233 and 71471167).

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