Interactive α -satisfactory method for multi-objective optimization with fuzzy parameters and linguistic preference

Abstract

An interactive α-satisfactory method via relaxed order of desirable α-satisfactory degrees is proposed for multi-objective optimization with fuzzy parameters and linguistic preference in this paper. Fuzzy parameters existing in objectives and constraints of multi-objective optimization are defined as fuzzy numbers and α-level set is used to build the feasible domain of parameters. On the basis, the original problem with fuzzy parameters is transformed into multi-objective optimization with fuzzy goals. Linguistic preference of decision-maker is modelled by the relaxed order of desirable α-satisfactory degrees of all the objectives. In order to achieve a compromise between optimization and preference, the multi-objective optimization problem is divided into two single-objective sub-problems: the preliminary optimization and the linguistic preference optimization. A preferred solution can be found by parameter adjustment of inner-outer loop. The minimum stable relaxation algorithm of parameter is developed for calculating the relaxation bound of maximum desirable satisfaction difference. The M-α-Pareto optimality of solution is guaranteed by the test model. The effectiveness, flexibility and sensitivity of the proposed method are well demonstrated by numerical example and application example to heat conduction system.

Keywords

Multi-objective optimization linguistic preference fuzzy parameter satisfactory degree

1 Introduction

In industry, production management decision-making, and national defense, the optimization of multiple objectives, i.e. multi-objective optimization(MOO), is commonly involved. For the feasible region of constraints, it is necessary to select the most satisfactory decision in terms of intention of decision-maker. In reality, multiple objectives are commonly conflicting, incommensurable, and even uncertain. Hence solving the MOO becomes an iterative decision-making process. It is very difficult to optimize all objectives. In order to seek a most preferred solution, a wide variety of methods and techniques have been developed [1 –7].

In MOO problems, objectives and constraints generally contain many parameters given by experiments or subjective judgments of decision-maker. However, in the real world, these parameters are often implicit due to decision-maker’s insufficient understanding to the system. For example, in economic application, some parameters are not clear, such as the investment return being about 2000 dollars. Possibly, decision-maker can only give the range of some coefficients of objectives and constraints. These coefficients are often uncertain but bounded. In 1970, Bellman and Zadeh [8] proposed the basic model of fuzzy decision-making. All the parameters, concepts and events that cannot be precisely defined by decision-maker are transformed into fuzzy sets containing a series of possible choices with different confidence levels. Hence it is more appropriate to regard these fuzzy parameter as fuzzy number. Diverse fuzzy optimization problems and fuzzy solutions are founded. Many scholars have proposed different interactive methods. The typical interactive satisfactory optimization method was proposed by Sakawa et al. [9]. Mohan and Nguyen [10] improved the algorithm further. Jafarian et al. [11] proposed a novel method integrating the concepts of intuitionistic fuzzy sets, interactive decision-making and geometric programming. Kannan et al. [12] combined fuzzy mathematical programming with multi-objective particle swarm optimization to deal with uncertain parameters and obtain solutions on Pareto frontier. To solve combinatorial MOO problems with fuzzy data, Bahri et al. [13] defined a fuzzy Pareto dominance for ranking the generated fuzzy solutions. Maysam [14] designed fuzzy adaptive multi-objective cat swarm optimization algorithm to estimate the Pareto frontier, where its parameters are tuned to new environment by Mamdani fuzzy rules when a change occurs.

In order to select specific results from Pareto solution set, preference information given by decision-maker plays an important role. A lot of research has been done in this field [15 –18]. Different preference information may produce different Pareto optimal solutions. In light of the expressions of preference information proposed by Huang and Masud in 1979, MOO problems can be divided into three categories: prior preference, interactive preference and posteriori preference [19].

Prior preference means that decision-maker can present adequate preference understanding for the optimization problem in advance. New optimization evaluation conditions or optimization rules can be built in light of preference. This kind of preference is usually characterized by the relative importance or priority requirement of the goal. Base on p-norm formulation, evaluation of objective functions can be transformed into a scalar criterion using weights to express preferrence. The weighted additive model is equivalent to 1-norm, while the weighted min-max approaches are on ∞-norm. Li et al. [20] developed a systematic way to incorporate decision-maker’s preference information into the decomposition-based evolutionary MOO methods. Thiele et al. [21] proposed a preference-based evolutionary approach being used as an integral part of interactive algorithm. Interactive preference means that the preference is not explicit. Due to the complexity of the problem, decision-maker does not know the optimization fully and has to adjust the preference information gradually in terms of the progressive results of optimization. Therefore the objectives’ preference becomes gradually clear until the optimal solution is obtained. The interactive techniques allow the solution to progress toward a preferred solution through an adaptive process during which the decision-maker’s preference is progressively elicited. Zionts et al. [22] proposed a human-machine interactive mathematical programming method to solve the multiple criteria problem. Liu et al. [23] developed a novel multi-objective evolutionary algorithm based on multi-layer interaction preference through decomposition. Posteriori preference means that the MOO problem is solved without preference firstly, and afterwards the preferred result is selected in light of decision-maker’s intention. In this case, it is very important to generate a large number of Pareto optimal solutions. It is required that all the solutions satisfy the Pareto optimality. Cheng et al. [24] defined a reference vector-guided evolutionary algorithm to decompose the original MOO into a number of single-objective sub-problems, and elucidate preference information for a preferred subset of the whole Pareto frontier. Wang et al. [25] proposed a improved two-archive algorithm to assign two selection principles (indicator-based and Pareto-based) to the two archives.

In all the preferences, the importance requirement of objectives is a kind of typical prior preference. Generally the relative importance can be expressed quantitatively, and the weighted model is conventional approach [26 –30]. The weight coefficients reflect the importance extents of objectives. The more important the objective is, the greater the weight coefficient value is. In actual production and decision-making, it is more convenient for decision-maker to express preference information in linguistic form. Many linguistic preference measurement methods have been developed in recent years. For example, Herrera et al. [31] expressed linguistic information by means of 2-tuples. Xu et al. [32, 33] focus on deviation measures of linguistic of linguistic preference relations and proposed virtual linguistic terms based on the syntax and semantics. Wang et al. [34] presents a novel linguistic representational and computational model in which the linguistic terms are weakened hedges. Gou et al. [35] defined a double hierarchy linguistic term set to express complex linguistic information. On the basis, they defined the concept of double hierarchy linguistic preference relation to reflect the relationships of any two alternatives and further proposed self-confident double hierarchy linguistic preference relation [36, 37]. Most models mentioned above are based of linguistic term set (LTS) denoted by S = {α|α = 0, 1, . . . , τ}, and the semantics of each S_α ∈ S is presented by a fuzzy membership function. These methods are wildly used in group decision making. However, in this paper, we mainly focus on the fuzzy importance toward multiple objectives with linguistic preference by only one decision-maker. Narasimhan [38] used language values such as “important”, “somewhat important”, “very important” to describe importance. And he defined the membership function for the fuzzy importance. Chen and Tsai [39] used the expected satisfaction to represent the fuzzy importance of objectives, and proposed the idea that the more important objective should have the higher expected satisfaction. A two-phase interactive satisfying optimization method was developed for fuzzy multiple objectives optimization with linguistic preference [40].

In this paper, fuzzy parameters and fuzzy importance defined by linguistic preference are both addressed in MOO. The fuzzy parameters are represented as fuzzy numbers. Correspondingly fuzzy systems are redefined by α-level set, so that the fuzzy parameters are used as variables to participate in the optimization. The relaxed order of desirable α-satisfactory degrees is defined to denote different linguistic preference. The original MOO model is transformed into two sub-problems, namely, preliminary optimization model and linguistic preference model. By solving the preliminary optimization model, the maximum comprehensive satisfaction λ^* is obtained, and then optimization and importance preference are balanced by the slack parameter Δδ in the preference model. When decision-maker is not satisfied with the optimization result, Δδ can be increased further to relax the constraint. When the adjustment of Δδ still does not meet the requirement of decision-maker, the confidence of fuzzy parameters α can be decreased to obtain a larger feasible region. On the basis, constraint confidence α and slack parameter Δδ are combined to form an interactive satisfactory optimization algorithm with inner-outer loop.Through the inner-outer loop interaction, the freedom degree of optimization is increased, and the more flexible and satisfactory solution can be found. Moreover, the minimum stable relaxation algorithm of Δδ is developed to calculate its relaxation bound corresponding to maximum desirable satisfaction difference. The M-α-Pareto optimality test model is design to guarantee the optimality of solution.

The innovations of this paper can be summarized as:

(1) In order to achieve the balance between optimization and preference, the MOO problem with preference is converted into two single-objective sub-problems including the preliminary optimization and the linguistic preference optimization.

(2) The relaxed order of desirable α-satisfactory degree is used to model the linguistic preference of decision-maker. Two freedom degrees of parameter adjustment resulting from inner-outer loop is built to improve the satisfaction of solution.

(3) Since the importance difference cannot be enlarged, the minimum stable relaxation algorithm is designed to obtain the relaxation bound.

In this paper, Section 2 describes MOO problem with fuzzy parameters and linguistic preferences. The interactive α-satisfactory algorithm with inner-outer loop is introduced in Section 3. Section 4 demonstrates its power by the numerical example and application. In Section 5 the conclusions are drawn.

2 MOO problem with fuzzy parameters and linguistic preference

2.1 MOO problem with fuzzy parameters

Correspondingly the MOO problem with fuzzy parameters can be expressed as follows: $\begin{matrix} min (f_{1} (x, {\tilde{a}}_{1}), . . ., f_{k} (x, {\tilde{a}}_{k})) \\ s . t . x \in G (\tilde{b}) = {x | g_{j} (x, {\tilde{b}}_{j}) \leq 0, j = 1, . . ., m} \end{matrix}$ (1) Where f_i (x) (i = 1, . . . , k) are objective functions. ${\tilde{a}}_{i} = ({\tilde{a}}_{i 1}, {\tilde{a}}_{i 2}, . . ., {\tilde{a}}_{i r_{i}})$ and ${\tilde{b}}_{j} = ({\tilde{b}}_{j 1}, {\tilde{b}}_{j 2}, . . ., {\tilde{b}}_{j s_{j}})$ are the fuzzy coefficient vectors of the objectives and the constraints respectively. They reflect the subjective ambiguity or objective uncertainty of parameters. In this paper, the fuzzy parameter is taken as fuzzy number, which is a continuous convex fuzzy subset in the real number field R. Its membership function is a continuous mapping of monotonically increasing or decreasing. In order to reduce calculation burden, some linear fuzzy numbers are used, such as triangles and trapezoidal fuzzy numbers, shown as Figure 1 and Figure 2 respectively. In this paper, the trapezoidal formulation is used to describe fuzzy parameter, and has the following membership function. $μ_{\tilde{c}} (c) = {\begin{matrix} 1 - \frac{c_{2} - c}{c_{2} - c_{1}} c_{1} \leq c \leq c_{2} \\ 1 c_{2} \leq c \leq c_{3} \\ 1 - \frac{c - c_{3}}{c_{4} - c_{3}} c_{3} \leq c \leq c_{4} \\ 0 otherwise \end{matrix}$ (2)

Fig.1

The triangular fuzzy number.

Fig.2

The trapezoidal fuzzy number.

It can be known that the α-level set of the fuzzy number ${\tilde{c}}_{α}$ is actually a closed interval on the real number field R, defined as ${\tilde{c}}_{α} = {c \in R | μ_{\tilde{c}} (c) \geq α} = [{c_{α}}^{L}, {c_{α}}^{R}]$ (3)

For the fuzzy parameters $\tilde{a}$ and $\tilde{b}$ , combined with the concept of α-level set, the following constraint inequality condition about the coefficient $(\tilde{a}, \tilde{b})$ can be formulated according to the actual demand or the expection of decision-maker. $(\tilde{a}, \tilde{b})_{α} = {(a, b) | \begin{matrix} μ_{{\tilde{a}}_{id}} (a_{id}) \geq α, μ_{{\tilde{b}}_{je}} (b_{je}) \geq α \\ i = 1, 2, . . ., k; d = 1, 2, . . ., r_{i} \\ j = 1, 2, . . ., m; e = 1, 2, . . ., s_{j} \end{matrix}$ (4)

The value of α corresponds to the size of feasible domain. When α = 1, the constraint is the most accurate, but the feasible range is the smallest. With the decrease of α, the constraint satisfaction becomes smaller, but the feasible domain gradually increases. In the case of α = 0, the constraint confidence is the weakest and the feasible range is the largest.

In the MOO, there are often strong conflicts among multiple goals. The conflict among multiple objectives may lead to the unsatisfactory optimization result. In order to reduce this conflict, decision-maker can assign the expectation value and tolerance degree for each objective in advance and convert the objective optimization into the goal satisfaction. The decision goal composed of expectation and tolerance is called fuzzy goal. Fuzzy goal is one kind of vague requirement of decision-maker. Consequently, the multi-objective optimization problem with fuzzy parameters can be transformed into a α-based MOO problem with fuzzy goals. $\begin{matrix} find (x, a) \\ such that f_{i} (x, a_{i}) \to f_{i}^{*}, i = 1, . . ., k \\ s . t . x \in G (\tilde{b}) = {x | g_{j} (x, {\tilde{b}}_{j}) \leq 0, j = 1, . . ., m} \\ (a, b) \in (\tilde{a}, \tilde{b})_{α} \end{matrix}$ (5) Where $f_{i}^{*}$ is the goal value of the i-th objective. “→” represents three types of fuzzy relationships, that is “ $\tilde{\leq}$ ”, “ $\tilde{\geq}$ ” and “ $\tilde{=}$ ”, denoting fuzzy minimization, fuzzy maximization and fuzzy equality.

“ $\tilde{\leq}$ ” indicates that the value of objective function is required to be approximately less than or equal to its perspective value $f_{i}^{*}$ . This means that decision-maker allows the objective value to be greater than $f_{i}^{*}$ in the range of tolerance $(f_{i}^{*}, {f_{i}}^{max})$ . f_i^max is the tolerance limit value, and the corresponding membership function is $μ_{f_{i}} (x, a_{i}) = {\begin{matrix} 1 f_{i} (x, a_{i}) \leq f_{i}^{*} \\ 1 - \frac{f_{i} (x, a_{i}) - f_{i}^{*}}{{f_{i}}^{max} - f_{i}^{*}} f_{i}^{*} \leq f_{i} (x, a_{i}) \leq {f_{i}}^{max} \\ 0 f_{i} (x, a_{i}) \geq {f_{i}}^{max} \end{matrix}$ (6) Its structure is shown in Figure 3.

Fig.3

Membership function μ_{f
_i} (x, a_i) for fuzzy relation “ $\tilde{\leq}$ ”.

The fuzzy relation “ $\tilde{\geq}$ ” indicates that the objective value is approximately greater than or equal to its perspective value. The tolerance interval is $({f_{i}}^{min}, f_{i}^{*})$ , and the tolerance limit is f_i^min. Then the membership function is written as: $μ_{f_{i}} (x, a_{i}) = {\begin{matrix} 1 f_{i} (x, a_{i}) \geq f_{i}^{*} \\ 1 - \frac{f_{i}^{*} - f_{i} (x, a_{i})}{f_{i}^{*} - {f_{i}}^{min}} {f_{i}}^{min} \leq f_{i} (x, a_{i}) \leq f_{i}^{*} \\ 0 f_{i} (x, a_{i}) \leq {f_{i}}^{min} \end{matrix}$ (7) Figure 4 shows its schematic.

Fig.4

Membership function μ_{f
_i} (x, a_i) for fuzzy relation “ $\tilde{\geq}$ ”.

“ $\tilde{=}$ ” denotes that the objective value is approximately equal to its goal (see Figure 5), which means that objective should get close to $f_{i}^{*}$ as much as possible. Its membership function is formulated as: $μ_{f_{i}} (x, a_{i}) = {\begin{matrix} 0 f_{i} (x, a_{i}) \geq {f_{i}}^{max} \\ \begin{matrix} 1 - \frac{f_{i} (x, a_{i}) - f_{i}^{*}}{{f_{i}}^{max} - f_{i}^{*}} f_{i}^{*} \leq f_{i} (x, a_{i}) \leq {f_{i}}^{max} \\ 1 f_{i} (x, a_{i}) = f_{i}^{*} \\ 1 - \frac{f_{i}^{*} - f_{i} (x, a_{i})}{f_{i}^{*} - {f_{i}}^{min}} {f_{i}}^{min} \leq f_{i} (x, a_{i}) \leq f_{i}^{*} \end{matrix} \\ 0 f_{i} (x, a_{i}) \leq {f_{i}}^{min} \end{matrix}$ (8)

Fig.5

Membership function μ_{f
_i} (x, a_i) for fuzzy relation “ $\tilde{=}$ ”.

When the objective function f_i is inferior to the tolerance limit, μ = 0. This means that the optimization result is completely unsatisfactory. When it reaches or exceeds $f_{i}^{*}$ , μ = 1. That is the optimization result conforms to the intention of decision-maker completely, i.e. complete satisfaction. In addition, when the objective value is located in the tolerance interval, 0 < μ < 1, and the satisfaction of optimization depends on the judgment of decision-maker.

Combining the definitions of membership function, fuzzy multi-objective optimization model with α-level can be presented as: $\begin{matrix} max μ_{f_{1}} (x, a_{1}), μ_{f_{2}} (x, a_{2}), . . ., μ_{f_{k}} (x, a_{k}) \\ s . t . x \in G (\tilde{b}) = {x | g_{j} (x, {\tilde{b}}_{j}) \leq 0, j = 1, . . ., m} \\ (a, b) \in (\tilde{a}, \tilde{b})_{α} \end{matrix}$ (9)

For different α, different optimal solutions can be obtained. When α is fixed, a certain confidence level of fuzzy parameters is determined and the feasible region is fixed. In the feasible region, the objectives corresponding to each solution have their own membership value, which indicates satisfaction of decision-maker to the objective optimization. Hence, the definition of α-satisfactory degree is presented as:

Definition 1. (α-satisfactory degree) The value of the membership function, i.e. the goal satisfaction under a fixed α, is α-satisfactory degree.

By the introduction of α, the MOO problem with fuzzy parameters can be regarded as a typical parametric programming problem, where α provides a freedom degree of regulation. In this case, the analyzer gives the initial value of α in advance, and then continuously adjusts it according to the intention of decision-maker.

Simular to the traditional multi-objective optimal solution, the new definitions for the optimal solution of MOO problem with fuzzy parameters are presented as follows.

Definition 2. (M-α-Pareto optimal solution) A point x^* ∈ G (b^*), where $(a^{*}, b^{*}) \in (\tilde{a}, \tilde{b})_{α}$ , is M-α-Pareto optimal solution if and only if there does not exit another solution x ∈ G (b) and $(a^{*}, b^{*}) \in (\tilde{a}, \tilde{b})_{α}$ , such that μ_{f
_i} (x, a_i) ≥ μ_{f
_i} (x^*, a_i) for all i, (i = 1, 2, . . . , k) and μ_{f
_h} (x, a_h) > μ_{f
_h} (x^*, a_h) for at least one h, (h = 1, 2, . . . , k).

Definition 3. (weak M-α-Pareto optimal solution) A point x^* ∈ G (b ^*), where $(a^{*}, b^{*}) \in (\tilde{a}, \tilde{b})_{α}$ , is weak M-α-Pareto optimal solution if and only if there does not exit another solution x ∈ G (b) and $(a^{*}, b^{*}) \in (\tilde{a}, \tilde{b})_{α}$ , such that μ_{f
_i} (x, a_i) > μ_{f
_i} (x^*, a_i) for all i, (i = 1, 2, . . . , k).

2.2 Linguistic preference

Prior preference is often characterized by its relative importance or priority. In order to reflect the difference of relative importance between objectives, weight coefficient ω_i, (i = 1, . . . , k) is used to build the evaluation form z_i (ω_i, f_i (x)) of each objective. $F = Z [z_{1} (ω_{1}, f_{1} (x)), . . ., z_{k} (ω_{k}, f_{k} (x))]$ (10) MOO problem is transformed into the single-objective optimization by (10). When the relative importance reaches the limit, it becomes a priority requirement. Supposing there are L priority levels, MOO problem is represented by the following model: $min [P_{1} (f_{1}^{1} (x), . . ., f_{l_{1}}^{1} (x)), . . ., P_{L} (f_{L}^{L} (x), . . ., f_{l_{L}}^{L} (x))]$ (11) where P_j is priority factor, indicating $f_{j}^{j} (x), . . ., f_{l_{j}}^{j} (x)$ belonging to j, j = 1, . . . , L priority level.

In light of optimization strategy, interactive methods can be roughly divided into interactive optimization method and interactive satisfaction method. The interactive optimization method requires decision-maker to give the trade-off ratio between the objectives according to the local information in the process of iterative calculation. The interactive satisfaction method requires decision-maker to judge the objectives that need to be improved, maintained and sacrificed in each iteration

For posteriori preference, in order to obtain the Pareto optimal solution set containing all the characteristics of optimization, the most commonly used methods are weighting method and ɛ-constraint method as (12) and (13). At present, some evolutionary algorithms are also developed to generate Pareto optimal frontier [24, 25]. $\begin{matrix} min \sum_{i = 1}^{k} ω_{i} f_{i} (x) \\ s . t . x \in G \end{matrix}$ (12)

$\begin{matrix} min f_{i} (x) \\ s . t . f_{j} (x) \leq ɛ_{j}, j = 1, . . ., k, j \neq i \\ x \in G \end{matrix}$ (13)

The importance of objectives is generally expressed by accurate weight information. In actual decision-making, however, due to the influence of environment and subjective understanding, decision-maker is not able to give accurate weights for all objectives. Moreover, the linguistic importance preference is more convenient for presentation of decision-maker’ intention. The fuzzy characteristics of linguistic information is more suitable for compromise of objectives.

In this paper, a unified language is used to describe the fuzzy importance preference. The following seven common linguistic terms are used to express different fuzzy importance information, that is

(1) “very important”

(2) “somewhat important”

(3) “important”

(4) “general”

(5) “unimportant”

(6) “somewhat unimportant”

(7) “very unimportant”

From (1) to (7), importance is decreasing. The above seven linguistic terms basically represent all the fuzzy description of importance.

3 Interactive α-Satisfactory method with inner-outer loop

3.1 Relaxed order of desirable α-Satisfactory degrees

For conventional importance preference, Chen and Tsai [39] defined the desirable satisfaction for each objective to present the relative importance. Hence the objective satisfaction should comply with $\begin{matrix} μ_{f_{i}} (x, a_{i}) \geq μ_{f_{i}}^{*} (x, a_{i}) \\ i \in {1, . . ., k}, (a, b) \in (\tilde{a}, \tilde{b})_{α} \end{matrix}$ (14) Where $μ_{f_{i}}^{*} (x, a_{i})$ represents the desirable α-satisfactory degree of the objective f_i. This requires that its actual optimal result is higher than the desirable α-satisfactory degree. However, decision-maker does not always understand nature of objective. Hence it is difficult to give accurate desirable α-satisfaction value.

In this paper, the desirable satisfaction is not constant, and treated as an optimization variable. The order of desirable α-satisfactory degree is established to present the importance difference of objectives. Such as the objective f_j is “very important”, and f_q is “somewhat important”, j, q ∈ { 1, . . . , k } , j ≠ q. The crisp comparison relation is formulated $μ_{f_{q}}^{*} (x, a_{q}) \leq μ_{f_{j}}^{*} (x, a_{j})$ (15)

The above formula only shows the desirable difference between relative importance.Nevertheless its comparative relationship is quite strict so that finding a more satisfactory solution becomes difficult. Therefore the difference variable γ is introduced to relax the desirable importance preference, so there is: $μ_{f_{q}}^{*} (x, a_{q}) - μ_{f_{j}}^{*} (x, a_{j}) \leq γ$ (16)

In this case, whether the desirable importance requirement is satisfied depends on inequality (16). When γ ≤ 0, the basic importance requirement (linguistic terms order) is conformed; on the contrary, when γ > 0, the fuzzy importance requirement is violated.

3.2 Interactive method

In order to balance objective optimization and importance difference, this paper decomposes the fuzzy multi-objective problem with linguistic preference into two sub-problems: preliminary optimization and linguistic preference optimization. Decision-maker is required to participate in optimization process.

3.2.1 Preliminary optimization

In this step, the purpose of preliminary optimization is to optimize all the objectives as much as possible regardless of linguistic preference. Once the structure of membership function is determined, the preliminary optimization model does not contain preference information. This guarantees that the satisfaction of the most conflicting objective is not too low. In this paper, the max-min formulation is utilized and the preliminary model is formulated as: $\begin{matrix} max λ \\ s . t . μ_{f_{i}} (x, a_{i}) \geq λ, i = 1, . . ., k \\ μ_{f_{i}} (x, a_{i}) \leq 1 \\ x \in G (\tilde{b}) = {x | g_{j} (x, {\tilde{b}}_{j}) \leq 0, j = 1, . . ., m} \\ (a, b) \in (\tilde{a}, \tilde{b})_{α} \end{matrix}$ (17) The optimal solution λ^* is called the maximum comprehensive satisfactory degree. It represents the maximum satisfaction that all the objectives even the worst objective can achieve the performance in the case of no preference. This problem is taken as the condition for the next step optimization.

3.2.2 Linguistic preference optimization

With the inequality (14), the feasible region of solution is reduced greatly. Especially when decision-maker pays special attention to a certain objective or some objectives, too high desriable satisfaction will be given. The preliminary optimization will make the feasible domain under linguistic preference become very small, even lead to no solution. So it may fail to meet the basic fuzzy importance requirement. Therefore, in order to obtain a satisfactory solution, it is necessary to give appropriate tolerance on the basis of the preliminary optimization model to expand the feasible region. The degree of relaxation depends on the intention of decision-maker. Consequently the following equation is proposed to relax the maximum comprehensive satisfactory degree λ^*. $μ_{f_{i}} (x, a_{i}) \geq μ_{f_{i}}^{*} (x, a_{i}) \geq λ^{*} - Δ δ, i = 1, . . ., k$ (18) Where Δδ is the slack parameter. The constraint (18) plays a key role in adjustment of the expected satisfaction $μ_{f_{i}}^{*} (x, a_{i})$ .

To satisfy the fuzzy importance requirement of multiple objectives, it is necessary to establish the second model, i.e. the linguistic preference model. Combining the desirable α-satisfactory relaxation constraint (18), there is $\begin{matrix} min γ \\ s . t . μ_{f_{i}} (x, a_{i}) \geq μ_{f_{i}}^{*} (x, a_{i}) \geq λ^{*} - Δ δ, i = 1, . . ., k \\ μ_{f_{s}}^{*} (x, a_{s}) - μ_{f_{t}}^{*} (x, a_{t}) \leq γ \\ membershipfunctions (6) or (7) or (8) \\ \begin{matrix} μ_{f_{i}} (x, a_{i}) \leq 1 \\ - 1 \leq γ \leq 1 \\ x \in G \end{matrix} \end{matrix}$ (19) In model (19), x, $μ_{f_{i}}^{*}$ , $μ_{f_{t}}^{*}$ , $μ_{f_{s}}^{*}$ , γ are all decision variables. The minimum γ means the maximum relative importance difference between objectives.

In this method, the slack parameter Δδ is used to control the feasible domain. Besides, the feasible domain can also be expanded by reducing α. When γ ≤ 0, the result satisfies the expected satisfaction order. That is different relaxation can correspond to different solutions.

3.2.3 Minimum stable relaxation algorithm

From the model (19), it can be seen that with the increase of Δδ, the constraints of real satisfaction and desirable satisfaction are relaxed. However, the desirable satisfaction difference between objectives will not increase any more. In this paper, the relaxation (Δδ^*) is called the minimum stable relaxation. $μ_{f}^{*}$ is used to express the vector including satisfactory degrees of all objectives, and the specific calculation process is as follows:

Step1: let Δδ=λ^*, then the optimal solution x^*, $μ_{f}^{*}$ , γ^* are obtained by solving the model (19).

Step2: find the lowest satisfactory degree in all objectives $μ_{f_{i}}^{*}$ , and let $Δ δ^{*} = λ^{*} - μ_{f_{i}}^{*}$ .

3.2.4 M-α-Pareto optimality test

The above model aims at achieving the desired α-satisfaction order, which may cause the actual optimal solution not to meet the M-α-Pareto characteristics. Hence it is also necessary to use the following M-α-Pareto test model to ensure the optimality. $\begin{matrix} max \sum_{i} ɛ_{i} \\ s . t . member ship functions (6) or (7) or (8) \\ μ_{f_{i}} (x, a_{i}) - ɛ_{i} = μ_{f_{i}} (x^{*}, a_{i}) \\ x \in G, ɛ_{i} \geq 0 \end{matrix}$ (20) Where x^* is the optimization solution of (19) and $\bar{x}$ is optimal solution of this model (20). The theorem of the M-α-Pareto optimality test is given as:

Theorem 1. If any component ɛ_i of ɛ is equal to zero, then x^* is M-α-Pareto optimal. If at least one ɛ_i is not zero, then $\bar{x}$ must be the M-α-Pareto optimal solution.

Proof. If x^* is not M-α-Pareto optimal when all ɛ_i are zero, $\bar{x}$ must be the M-α-Pareto optimal. Because $μ_{f_{i}} (\bar{x}, a_{i}) \geq μ_{f_{i}} (x^{*}, a_{i}), (i = 1, . . ., k)$ must hold, and $μ_{f_{h}} (\bar{x}, a_{i}) > μ_{f_{h}} (x^{*}, a_{i})$ for at least one h, h ∈ {1, . . . , k}. Then there must exist at least one non-zero ɛ_h, which is contrary to the fact that all ɛ_i are zero. So x^* is M-α-Pareto optimal when all ɛ_i are zero.

If at least one ɛ_i is not zero, then $\bar{x}$ must be the M-α-Pareto optimal solution. Otherwise, there will exist another M-α-Pareto optimal solution $\tilde{x}$ and it must be the optimal solution of (20), which is contrary to the fact that $\bar{x}$ is optimal solution. Therefor, $\bar{x}$ must be the M-α-Pareto optimal solution when at least one ɛ_i is not zero.

3.3 α-Satisfactory optimization structure with inner-outer loop

For the fuzzy multi-objective optimization problem with fuzzy parameters, its fuzziness is shown by the fuzzy coefficients in the objective and constraint functions. According to the Section 2, the fuzzy numbers can be used to describe the uncertain characteristics of fuzzy parameters. When linguistic preference information is encountered, it is still handled by fuzzy concept and an interactive α-satisfactory algorithm with Inner-outer loop can be formed.

In the proposed method, the constraint confidence degree α of fuzzy parameters corresponds to different size of constraint domains. To find the preferred solution, the optimization is constructed as inner-outer loop. The inner loop is used to regulate the tradeoff between the two sub-problems under a fixed α. The outer loop is a parametric programming layer of α. Optimization starts from the inner loop. If the optimization result of inner loop cannot meet the preference requirements of decision-maker, the value of α in outer loop can be changed to improve the result further. The range of α is 1-0, and its value is gradually adjusted from large to small.

3.4 Algorithms

Algorithm 1 Optimization Algorithms with Inner-outer Loop

Require: f (x): objective function; g (x): constraint; $\tilde{a}, \tilde{b}$ : fuzzy parameters;

Ensure: optimal x

1: Calculate $f_{i}^{max}$ and $f_{i}^{min}$ when α = 1 and α = 0;

2: Decision-maker chooses the objective desirable value and the initial tolerance to build the membership function μ_{f
_i};

3: Initialize α;

4: repeat

5: Establish the preliminary optimization model and get λ^*;

6: Initialize Δδ = 0 and formulate the relaxed order of desirable α-satisfactory degrees;

7: Establish the linguistic preference opimization model;

8: repeat

9: Substitute Δδ and solve the linguistic preference optimization model to get γ;

10: Test M-α-Pareto optimality of the solution and get x;

11: Increase Δδ by 0.05;

12: until optimization result meets the requirement or the value of Δδ is too large

13: Decrease α;

14: until the optimization result meets the requirement given by decision-maker

The following algorithm for MOO problem with fuzzy parameters and linguistic preference (as shown in Figure 6 and Algorithm 1):

Fig.6

Flow chart of algorithms.

Step 1. Calculate the maximum value $f_{i}^{max}$ and the minimum value $f_{i}^{min}$ of the objective function f_i (x, a_i) with the system constraint, when α = 1 and α = 0.

Step 2. According to the four optimal values of the individual objective, decision-maker gives the objective desirable value and the initial tolerance to formulate the membership function, and then gives the initial value to the confidence α.

Step 3. Establish the preliminary optimization model and compute the maximum comprehensive satisfaction λ^*.

Step 4. Assume Δδ = 0, build the relaxed order of desirable α-satisfactory degrees according to the linguistic preference, and establish the linguistic preference opimization model.

Step 5. Substitute λ^* and solve the linguistic preference optimization model.

Step 6. Test M-α-Pareto optimality of the solution by (20).

Step 7. Judge: if optimization result meets the importance difference and decision-maker is satisfied, the optimization stops; otherwise, go to step 8.

Step 8. Increase Δδ to further relax λ^*, then return to step 5; or decrease α, and then go back to step 3.

4 Simulation

The effectiveness of the proposed interactive α-satisfactory method with inner-outer loop is demonstrated by numerical example and application example to head conduction system.

4.1 Numerical example

The following numerical example generally originates from industry problems, e.g. production planning, where production cost is minimized and profit is maximized [9, 10]. Some parameters of the objective functions and constraints are imprecise. $\begin{matrix} {\begin{matrix} min f_{1} (x, {\tilde{a}}_{1}) = (x_{1} + 5)^{2} + {\tilde{a}}_{11} x_{2}^{2} + 2 (x_{3} - {\tilde{a}}_{12})^{2} \\ min f_{2} (x, {\tilde{a}}_{2}) = {\tilde{a}}_{21} (x_{1} - 45)^{2} + (x_{2} + 15)^{2} \\ + 3 (x_{3} + {\tilde{a}}_{22})^{2} \\ min f_{3} (x, {\tilde{a}}_{3}) = {\tilde{a}}_{31} (x_{1} + 20)^{2} + {\tilde{a}}_{32} (x_{2} - 45)^{2} \\ + (x_{3} + 15)^{2} \end{matrix} \\ g (x, {\tilde{b}}_{1}) = {\tilde{b}}_{11} x_{1}^{2} + {\tilde{b}}_{12} x_{2}^{2} + {\tilde{b}}_{13} x_{3}^{2} \leq 100 \\ 0 \leq x_{i} \leq 10, i = 1, 2, 3 \end{matrix}$ (21) Where $\tilde{a} = ({\tilde{a}}_{1}, {\tilde{a}}_{2}, {\tilde{a}}_{3})$ and $\tilde{b} = ({\tilde{b}}_{1}, {\tilde{b}}_{2})$ are fuzzy parameter vectors, and represented by the trapezoidal fuzzy numbers in Table 1.

Table 1

Fuzzy parameters

$\tilde{c}$	(c₁, c₂, c₃, c₄)	$\tilde{c}$	(c₁, c₂, c₃, c₄)
${\tilde{a}}_{11}$	(3.8, 4, 4, 4.3)	${\tilde{a}}_{32}$	(4.7, 5, 5, 5.35)
${\tilde{a}}_{12}$	(48.5, 50, 50, 52)	${\tilde{b}}_{11}$	(0.9, 1, 1, 1.1)
${\tilde{a}}_{21}$	(1.85, 2, 2, 2.2)	${\tilde{b}}_{12}$	(0.8, 1, 1, 1.2)
${\tilde{a}}_{22}$	(18.2, 20, 20, 22.5)	${\tilde{a}}_{13}$	(0.85, 1, 1, 1.15)
${\tilde{a}}_{31}$	(2.9, 3, 3, 3.15)

The linguistic preference requirement is $f_{1} (x, {\tilde{a}}_{1})$ is very important, $f_{2} (x, {\tilde{a}}_{2})$ is somewhat important, $f_{3} (x, {\tilde{a}}_{3})$ is important.

To show its power, the proposed method is implemented and compared with Chen’s method in [39].

Firstly, when α = 0 and α = 1, calculate the minimum and maximum values of each objective. Four optimal values of all objectives are shown in Table 2.

Table 2

Optimum of individual objective

	α = 0		α = 1
	maximum	minimum	maximum	minimum
$f_{1} (x, {\tilde{a}}_{1})$	2989.5	5932.4	3225	5433.3
$f_{2} (x, {\tilde{a}}_{2})$	3485	7997.4	3875	7002.9
$f_{3} (x, {\tilde{a}}_{3})$	7142.5	14008	7550	13078

As shown in Table 3, decision-maker takes the minimum and maximum of the four optimal values of each objective as the expected value and the allowable limit value.

Table 3

Perspective value and original tolerance of individual objective

	Perspective value	Tolerance
$f_{1} (x, {\tilde{a}}_{1})$	2989.5	5932.4
$f_{2} (x, {\tilde{a}}_{2})$	3485	7997.4
$f_{3} (x, {\tilde{a}}_{3})$	7142.5	14008

Therefore, the following membership functions are formed as: ${\begin{matrix} \begin{matrix} μ_{f_{1}} (x, a_{1}) = (5932.4 - (x_{1} + 5)^{2} - a_{11} x_{2}^{2} \\ - 2 (x_{3} - a_{12})^{2}) / 2942.9 \end{matrix} \\ \begin{matrix} μ_{f_{2}} (x, a_{2}) = (7997.4 - a_{21} (x_{1} - 45)^{2} \\ - (x_{2} + 15)^{2} - 3 (x_{3} + a_{22})^{2}) / 4512.4 \end{matrix} \\ \begin{matrix} μ_{f_{3}} (x, a_{3}) = (14008 - a_{31} (x_{1} + 20)^{2} \\ - a_{32} (x_{2} - 45)^{2} - (x_{3} + 15)^{2}) / 6865.5 \end{matrix} \end{matrix}$ (22)

Using the above equations, the preliminary optimization model is constructed and solved. Then λ^* = 0.6562 is obtained. The following linguistic preference optimization model is established based on the given linguistic preference. $\begin{matrix} min γ \\ s . t . μ_{f_{i}} (x, a_{i}) \geq μ_{f_{i}}^{*} \geq λ^{*} - Δ δ, i = 1, . . ., 3 \\ \begin{matrix} μ_{f_{2}}^{*} - μ_{f_{1}}^{*} \leq γ \\ μ_{f_{3}}^{*} - μ_{f_{2}}^{*} \leq γ \end{matrix} \\ μ_{f_{i}} (x, a_{i}) \leq 1 \\ \begin{matrix} - 1 \leq γ \leq 1 \\ g (x, {\tilde{b}}_{1}) = {\tilde{b}}_{11} x_{1}^{2} + {\tilde{b}}_{12} x_{2}^{2} + {\tilde{b}}_{13} x_{3}^{2} \leq 100 \\ 0 \leq x_{i} \leq 10, i = 1, 2, 3 \\ 2989.5 \leq f_{1} \leq 5932.4 \\ 3485 \leq f_{2} \leq 7997.4 \\ 7142.5 \leq f_{3} \leq 14008 \\ (a, b) \in (\tilde{a}, \tilde{b})_{α} \end{matrix} \end{matrix}$ (23)

Initialize α = 0.9 and Δδ = 0, then the result is shown in Table 4. Obviously, the result λ^* needs to be relaxed by properly increasing Δδ. Then reformulate (23) and solve it again. Further, α is decreased to improve the optimization result. The solutions are listed in Table 4. When α = 0.9 and Δδ = 0.1282, the optimization results of all objectives are consistent with decision-maker’s linguistic preference. Refer to the approach proposed by Chen and Tsai [39], and suppose the desirable satisfactory degrees of the three fuzzy goals are (0.6, 0.5, 0.4), and the satisfactory degrees are obtained as (0.6000, 0.5000, 0.6773) when α = 0.9 and (0.6112, 0.5000, 0.7309) when α = 0.7.

Table 4

Optimization results for original linguistic preference

α	λ ^*	Δδ	γ	Desirable α-satisfactory degree	α-satisfactory degree	Objective value
0.9	0.5782	0	-0.0036	(0.5854, 0.5818, 0.5782)	(0.5854, 0.5818, 0.5835)	(4209.6, 5372.1, 10002.2)
		0.0782	-0.0612	(0.6225, 0.5612, 0.5000)	(0.6225, 0.5612, 0.5663)	(4100.5, 5464.8, 10120)
		0.1282	-0.1034	(0.6569, 0.5534, 0.4500)	(0.6569, 0.5534, 0.5286)	(3999.3, 5500.1, 10378.9)
0.7	0.5970	0	-0.0028	(0.6026, 0.5998, 0.5970)	(0.6026, 0.5998, 0.6122)	(4159, 5290.9, 9804.9)
		0.0970	-0.0736	(0.6471, 0.5736, 0.5000)	(0.6471, 0.5736, 0.5947)	(4028, 5409.3, 9925)
		0.1470	-0.1151	(0.6802, 0.5651, 0.4500)	(0.6802, 0.5651, 0.5609)	(3930.6, 5447.4, 10157.2)
		0.1970	-0.1555	(0.7109, 0.5555, 0.4000)	(0.7109, 0.5555, 0.5283)	(3840.2, 5491, 103807)

The 2D and 3D Pareto frontier of the proposed method can be seen in Figure 7 when α = 0.7 and α = 0.9.

Fig.7

Pareto frontier.

4.1.1 Analysis of minimum stable relaxation

Let Δδ=λ^*, and solve the model (23), γ^* = -0.2570 is obtained and the satisfactory degrees of all objectives are (0.6065, 0.4542, 0.7203). According to the lowest satisfaction $μ_{f_{i}}^{*} = 0.4542$ , the minimum stable relaxation Δδ^* = 0.1240 can be calculated.

4.1.2 Efficiency

If decision-maker puts forward new linguistic preference requirement for MOO: $f_{1} (x, {\tilde{a}}_{1})$ is very important, $f_{2} (x, {\tilde{a}}_{2})$ is somewhat important, $f_{3} (x, {\tilde{a}}_{3})$ is unimportant. Although the importance of objective $f_{3} (x, {\tilde{a}}_{3})$ has changed, the order of linguistic values and the minimum difference of preference between objectives have not changed. Therefore, directly selecting α = 0.9 and Δδ = 0.1282 from the Table 4 is still satisfactory. The same optimal results by Chen’s approach can be obtained when the desirable satisfactory degrees are (0.6, 0.5, 0.2).

4.1.3 Flexibility

Flexibility needs to be verified by dealing with different degrees of preference (the minimum difference between the desirable satisfaction). Assume that decision-maker does not change the order of linguistic values, but just change the degree of preference, e.g. $f_{1} (x, {\tilde{a}}_{1})$ is very important, $f_{2} (x, {\tilde{a}}_{2})$ is important, $f_{3} (x, {\tilde{a}}_{3})$ is unimportant. α = 0.9 and Δδ = 0.1282 in the Table 4 are just suitable for the situation where the importance preference difference is relatively small. Consequently, decrease α to 0.7 and increase Δδ from 0, and then get the results. It can be seen that when α = 0.7 and Δδ = 0.1970, γ = -0.1555, the degree of preference has increased, and the difference between the desirable satisfaction is consistent with the preference of the new linguistic value. Using Chen’s method, it is assumed that the desirable satisfactory degrees for the goals are (0.6, 0.4, 0.2). Then the satisfactory degrees are (0.6065, 0.4542, 0.7203) when α = 0.9 and (0.6321, 0.4712, 0.7398) when α = 0.7.

Comparing the results of these two methods, it can be found that the method proposed by this paper is better than Chen’s. The reason is that Chen’s method is actually single one fixed optimization in the inner loop. It can only give the desire of satisfactory degrees in terms of preference for each objective. However, the adjustment of inner loop in this paper is interactive. Hence, when the optimization results cannot meet the preference well, the preference degree can be enlarged by relaxing the constraints by outer-loop.

4.1.4 Sensitivity

The sensitivity of this algorithm is validated when the order of linguistic values changes. It is assumed that the new linguistic preference is: $f_{1} (x, {\tilde{a}}_{1})$ is very important, $f_{2} (x, {\tilde{a}}_{2})$ is unimportant, and $f_{3} (x, {\tilde{a}}_{3})$ is important. Correspondingly, the linguistic preference optimization is reformulated as follows: $\begin{matrix} min γ \\ s . t . μ_{f_{i}} (x, {\tilde{a}}_{i}) \geq μ_{f_{i}}^{*} \geq λ^{*} - Δ δ, i = 1, . . ., 3 \\ \begin{matrix} μ_{f_{3}}^{*} - μ_{f_{1}}^{*} \leq γ \\ μ_{f_{2}}^{*} - μ_{f_{3}}^{*} \leq γ \end{matrix} \\ μ_{f_{i}} (x, {\tilde{a}}_{i}) \leq 1 \\ \begin{matrix} - 1 \leq γ \leq 1 \\ g (x, {\tilde{b}}_{1}) = {\tilde{b}}_{11} x_{1}^{2} + {\tilde{b}}_{12} x_{2}^{2} + {\tilde{b}}_{13} x_{3}^{2} \leq 100 \\ 0 \leq x_{i} \leq 10, i = 1, 2, 3 \\ 2989.5 \leq f_{1} \leq 5932.4 \\ 3485 \leq f_{2} \leq 7997.4 \\ 7142.5 \leq f_{3} \leq 14008 \\ (a, b) \in (\tilde{a}, \tilde{b})_{α} \end{matrix} \end{matrix}$ (24)

Solve (24) and get Table 5. When γ = -0.0863, the result meets the requirement.

Table 5

Optimization results of sensitivity

α	λ ^*	Δδ	γ	Desirable α-satisfactory degree	α-satisfactory degree	Objective value
0.9	0.5782	0	-0.0026	(0.5834, 0.5782, 0.5808)	(0.5834, 0.5782, 0.5915)	(4215.4, 5388.3, 9946.8)
		0.0782	-0.0718	(0.6436, 0.5000, 0.5718)	(0.6436, 0.5000, 0.6266)	(4038.4, 5741.2, 9706.1)
0.7	0.5970	0	-0.0035	(0.6039, 0.5970, 0.6004)	(0.6039, 0.5970, 0.6149)	(4155.2, 5303.5, 9786.6)
		0.0970	-0.0863	(0.6727, 0.5000, 0.5863)	(0.6727, 0.5000, 0.6634)	(3952.7, 5741.2, 9453.6)

4.2 Application example

To illustrate the proposed method, a four objective linear optimal control problem of one dimensional heat conduction to control temperature distribution is used as test example. The simulation result is compared with LTS-based model [31]. A LTS with 7 linguistic term of importance defined in the interval in [0,1] is shown in Fig. 8.

Fig.8

LTS of importance.

A multi-objective optimal control problem in a linear distributed-parameter system governed by the process of one-sided heating of metal in a furnace is considered [41, 42]. It is described by the diffusion equation $\frac{\partial^{2} q (y, t)}{\partial y^{2}} = \frac{\partial q (y, t)}{\partial t}$ (25) where q (y, t) is the temperature distribution in the metal depending on the space coordinate y (0 ≤ y ≤ 1) and the currently heating time t (0 ≤ t ≤ T).

The initial condition about heating time and boundary conditions about space coordinate are given by $q (y, 0) = 0$ (26) ${\frac{\partial q (y, t)}{\partial y} |}_{y = 0} = η {q (0, t) - ν (t)}$ (27) ${\frac{\partial q (y, t)}{\partial y} |}_{y = 1} = 0$ (28) where η is the heat transfer coefficient. v (t) is the temperature of the gas medium and it is controlled by the fuel flow u (t) (0 ≤ u (t) ≤1). There is $ξ \frac{d ν (t)}{dt} + ν (t) = u (t)$ (29) where ξ is the time constant of the furnace.

The temperature distribution q (y, t) can be calculated for a given control function u (t) with the function g (y, t), which is shown as follow: $q (x, t) = \int_{0}^{t} g (y, t - τ) u (τ) d τ$ (30)

$\begin{matrix} g (y, t) = \frac{κ^{2} cos (1 - y)}{cos κ - (κ / η) sin κ} e^{- κ^{2} t} \\ + 2 κ^{2} \sum_{i = 0}^{\infty} \frac{cos (1 - y) β_{i}}{(κ^{2} - {β_{i}}^{2}) (\frac{1}{η} + \frac{1 + η}{{β_{i}}^{2}}) cos β_{i}} e^{- {β_{i}}^{2} t} \end{matrix}$ (31) where $κ = 1 / \sqrt{ξ}$ , $β_{i}^{'}$ s are the real roots of the transcendental equation β tan β = η.

Some of the most popular performance indices are used for linear distributed-parameter systems on the basis of saving the energy as far as possible.

(1) Minimum total fuel. $J_{1} = \int_{0}^{T} u (t) dt$ (32)

(2) Minimum amplitude of the fuel flow. $J_{2} = max_{0 \leq t \leq T} u (t)$ (33)

(3) Minimum sum of temperature error in the whole space. $J_{3} = \int_{0}^{1} | q^{*} - q (y, t) | dy$ (34)

(4) Minimum temperature error of any space coordinate. $J_{4} = max_{0 \leq y \leq 1} | q^{*} - q (y, t) |$ (35)

q^* is the expected spatial temperature distribution. Correspondingly a MOO problem is formulated. In order to simplify this optimal control problem, the continuous objectives are discretized by means of Simpson’s numerical integration formula. Therefore, the MOO model with linguistic preference can be written as follows: $\begin{matrix} min (f_{1} (x), f_{2} (x), f_{3} (x), f_{4} (x)) \\ s . t . q^{*} (y_{i}) - T \sum_{h = 0}^{H} d_{h} g (y_{i}, T - τ_{h}, η, ξ) u (τ_{h}) \\ = q_{j}^{+} + q_{j}^{-} \\ q_{j}^{+} + q_{j}^{-} \leq Q, q_{j}^{+} \geq 0, q_{j}^{-} \geq 0, j = 0, . . ., N \\ u (τ_{h}) \leq U, 0 \leq u (τ_{h}) \leq 1, h = 0, . . ., H \\ (η, ξ) \in (\tilde{η}, \tilde{ξ})_{α} \end{matrix}$ (36) where $x = (u (τ_{0}), . . ., u (τ_{h}) . q_{0}^{+}, q_{0}^{-}, . . ., q_{N}^{+}, q_{N}^{-}, Q,$ U), η and ξ are regarded as fuzzy parameters in this system, and ${\begin{matrix} f_{1} (x) = T \sum_{h = 0}^{H} c_{h} u (τ_{h}) \\ f_{2} (x) = U \\ f_{3} (x) = \sum_{j = 0}^{N} d_{j} (q_{j}^{+} + q_{j}^{-}) \\ f_{4} (x) = Q \end{matrix}$ (37)

Here, decision-maker gives the linguistic preference in light of real requirement of industry:

f₁ (x) is important;

f₂ (x) is important;

f₃ (x) is very important;

f₄ (x) is very important.

Let H = 20, N = 20, $\tilde{η} = (8, 10, 10, 12)$ and $\tilde{ξ} = (0.03, 0.04, 0.04, 0.05)$ . We use the proposed method to solve this problem. Firstly, calculate the minimum and maximum values of each objective when α = 0 and α = 1. The proper perspective values and tolerant limits of the objectives are (0.0000, 0.1502), (0.0000, 1.0000), (0.0026, 0.2000), (0.0059, 0.2000) in this paper. Thus, their membership functions are ${\begin{matrix} μ_{f_{1}} (x) = (0.1502 - T \sum_{h = 0}^{H} c_{h} u (τ_{h})) / 0.1502 \\ μ_{f_{2}} (x) = (1 - U) / U \\ μ_{f_{3}} (x) = (0.2 - \sum_{j = 0}^{N} d_{j} (q_{j}^{+} + q_{j}^{-})) / 0.1974 \\ μ_{f_{4}} (x) = (0.2 - Q) / 0.1941 \end{matrix}$ (38)

The preliminary optimization model is built and solved. Then λ^* is obtained. Hence, the linguistic preference optimization model is $\begin{matrix} min γ \\ {s . t . μ_{f_{i}} (x, a_{i}) \geq μ_{f_{i}}}^{*} \geq λ^{*} - Δ δ, i = 1, . . ., 3 \end{matrix} \begin{matrix} μ_{f_{1}}^{*} - μ_{f_{3}}^{*} \leq γ \\ μ_{f_{2}}^{*} - μ_{f_{3}}^{*} \leq γ \\ μ_{f_{1}}^{*} - μ_{f_{4}}^{*} \leq γ \\ μ_{f_{2}}^{*} - μ_{f_{4}}^{*} \leq γ \end{matrix}$ $\begin{matrix} μ_{f_{i}} (x, a_{i}) \leq 1 \\ \begin{matrix} - 1 \leq γ \leq 1 \\ membershipfunctions (38) \\ q^{*} (y_{i}) - T \sum_{h = 0}^{H} d_{h} g (y_{i}, T - τ_{h}, η, ξ) u (τ_{h}) \\ = q_{j}^{+} + q_{j}^{-} \\ q_{j}^{+} + q_{j}^{-} \leq Q, q_{j}^{+} \geq 0, q_{j}^{-} \geq 0, j = 0, . . ., N \\ u (τ_{h}) \leq U, 0 \leq u (τ_{h}) \leq 1, h = 0, . . ., H \\ (η, ξ) \in (\tilde{η}, \tilde{ξ})_{α} \end{matrix} \end{matrix}$ (39)

The corresponding satisfactory degrees in the optimization results of (39) for different α and Δδ are given to reflect the various decision making requirements in Table 6. And the satisfactory degrees of LTS-based method are (0.2445,0.5126,0.8415,0.6278) when α = 0.5 and (0.261,0.5259,0.831,0.6069) when α = 0.9.

Table 6

Optimization results for application example

α	λ ^*	Δδ	γ	Desirable α-satisfactory degree	α-satisfactory degree
0.9	0.4855	0.0855	-0.1951	(0.4000, 0.4000, 0.5951, 0.5951)	(0.4009, 0.4314, 0.6246, 0.5896)
		0.2855	-0.5689	(0.2000, 0.2000, 0.7688, 0.7688)	(0.2027, 0.3833, 0.8577, 0.7616)
		0.4055	-0.8398	(0.0800, 0.0800, 0.9198, 0.9198)	(0.0805, 0.1285, 0.9417, 0.9203)
0.5	0.4871	0.2871	-0.5797	(0.2000, 0.2000, 0.7797, 0.7797)	(0.2161, 0.3988, 0.8648, 0.7849)
		0.4071	-0.8166	(0.0800, 0.0800, 0.8966, 0.8966)	(0.0870, 0.2747, 0.9455, 0.8789)

Figs. 9 and 10 are the curves of the temperature distribution q (y, T) and the control function u (t) when α = 0.5 and Δδ = 0.4071 by these two method. The results of application further prove the effectiveness of the proposed method.

Fig.9

Temperature distribution q (y, T).

Fig.10

Flow control function u (t).

Comparing the results of the two methods, it can be seen that the method proposed by this paper is better than LTS-based method. The results of the proposed method present the fuzzy importance more reasonably. In LTS-based method, once the linguistic preference is given by decision-maker, the optimization result is fixed. However, decision-maker can balance optimization and importance through interactive parameter adjustment by the proposed method. Therefore, this method can express intention of decision-maker more reasonably and conveniently.

5 Conclusions

This paper presents an interactive α-satisfactory method for MOO problem with fuzzy parameters and linguistic preference. An inner-outer loop optimal adjustment structure is formulated with the α-level set and the slack parameter Δδ. In inner loop, two sum-problems are established by introducing the relaxation variable Δδ and the preference difference variable γ. The satisfactory solution is obtained by repeatedly adjusting the parameters and solving the models. The minimum stable relaxation algorithm is designed for relaxation bound and the M-α-Pareto optimality of solution is ensured by test model. The optimization results of the numerical example and application example to heat conduction system verify the effectiveness, flexibility and sensitivity of the algorithm.

Footnotes

Acknowledgments

This work is supported by National Natural Science Foundation of China under Grant (No. 61773279) and Key Technologies Program of Tianjin (19YFHBQY00040). The authors are grateful to the anonymous reviewers for their helpful comments and constructive suggestions with regard to this paper.

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