Multi-objective structural design problem optimization using parameterized t-norm based fuzzy optimization programming technique

Abstract

Real world engineering design problems are usually characterized by the presence of many conflicting objectives. In this paper, we will make an approach to solve multi-objective structural model using parameterized t-norms based on fuzzy optimization programming technique. Here, binary t-norms are expressed in extended n-ary t-norms and their basic properties and some special cases have been discussed. In this structural model formulation, the objective functions are the weight of the truss and the deflection of loaded joint; the design variables are the cross-sections of the truss members; the constraints are the stresses in members. A classical truss optimization example is presented here to demonstrate the efficiency of our proposed optimization approach. The test problem includes a three-bar planar truss subjected to a single load condition. This approximation approach is used to solve this multi-objective structural optimization model. The model is illustrated with numerical examples.

Keywords

Multi-objective optimization t-norms fuzzy set structural optimization

1 Introduction

This Optimization seeks to maximize the performance of a system, part or component, while satisfying design constraints. One common form of optimization is trial and error which is used every day. We make decisions, observe the result, and change future actions depending on the success of those decisions. When performing optimization, we wish to minimize (or maximize) the system response, while considering both design variables and design constraints. Design variables are variables the designer or engineer can freely choose from. For example, the thickness of a wall, the material chosen, and the width of a part. The resulting stress, deflection, volume, natural frequency and other typical performance measures are often considered either as objective functions or as constraints. Objective functions are the system responses that we wish to minimize, while constraints are limits that we impose on the system.

The common approach that constitutes this sort of problem is to choose of one objective and incorporate the other objectives as constraints. This approach is accompanied by its disadvantage of limiting the choices available for the designers, making the optimization process a rather difficult task.

Another common approach is to combine all objectives into a single objective function. This technique has the drawback of modeling the original problem in an inadequate manner, generating solutions that will require a further sensitivity analysis to become reasonably useful to the designer.

Different solutions may produce different trade-offs (conflicting scenarios) among different objectives. A solution that is extreme (in a better sense), with respect to one objective requires a compromise in other objectives. In multi-objective optimization, there may not exist a solution that is best with respect to all objectives. Instead, there are equally good solutions which are known as Pareto optimal solutions. A Pareto optimal set of solution is one such, that, when we go from any one point to another point in the set, at least one objective function improves and at least one other worsens. None of the solutions dominate over the other. All the sets of decision variables on the Pareto front are equally good and are expected to provide flexibility for the decision maker. Normally, the decision about “what the best answer is” corresponds to the so-called decision maker. Finally,the book by Eschenauer et al. [5] is a very valuable guide to some of the most relevant work in multi-objective design optimization in the last few years. Good surveys on structural optimization may be in literature [2 , 23].

In real life, the data cannot be recorded or collected precisely due to human errors or some unexpected situations. So,one may consider ambiguous situations like vague parameters, non-exact objective and constraint functions in the problem and it may be classified as a non-stochastic imprecise model. Here, fuzzy set theory may provide a method to describe or formulate this imprecise model. Zadeh [27] first gave the concept of fuzzy set theory for handling uncertainty that is due to imprecision rather than due to randomness. Later on Bellman and Zadeh [12] used the fuzzy set theory for the decision making problem. Zimmermann [28] proposed a fuzzy multi-criteria decision making set, defined as the intersection of all fuzzy goals and their constraints.

In practical, the problem of structural design may be formed as a typical non-linear programming problem with non-linear objective functions and constraints functions in fuzzy environment. Some researchers applied the fuzzy set theory for Structural model. For example Wang et al. [24] first applied α-cut method to structural designs where the non-linear problems were solved with various design levels α, and then a sequence of solutions are obtained by setting different level-cut value of α. Rao et al. [16, 17] showed significant work in multi-objective structural optimization with uncertain parameters. Rao [15] was one of the first to point out the importance of incorporating concepts of game theory into structural optimization and used several mathematical programming techniques to solve multi-objective structural optimization problems. Rao [18] applied the same α-cut method to design a four–bar mechanism for function generating problem. Structural optimization with fuzzy parameters was developed by Yeh et al. [26]. In 1989, Xu [25] used two-phase method for fuzzy optimization of structures. In 2004, Shih et al. [22] used level-cut approach of the first and second kind for structural design optimization problems with fuzzy resources.Shih et al. [21] develop an alternative α-level-cuts methods for optimum structural design with fuzzy resources in 2003. Dey et al. [3] optimize structural model in fuzzy environment.

Alsina et al. [1] introduced the t-norm in fuzzy set theory and suggested that the t-norms could be used for the intersection of fuzzy sets. Different types of t-norms theory and their fuzzy inference methods were introduced by Gupta et al. [4]. The extension of fuzzy implication operators and generalized fuzzy methods of cases were discussed by Ruan et al. [19]. Pei et al. [10] introduced the extended t-norms and another kind of fuzzy universal algebras. Kaymak et al. [18] use weighted extension of (Archimedean) fuzzy t-norms in optimization of various criteria model. Samanta et al. [20] solve portfolio selection model using extended t-norm based fuzzy optimization technique.

In this paper we are making an approach to solve multi-objective structural model using parameterized t-norms based fuzzy optimization programming technique. In this structural model formulation, the objective functions are (i) minimize weight of the truss and (ii) minimize deflection of loaded joint; the design variables are the cross-sections of the truss members; the constraints are the stresses in members. The test problem includes a three-bar planar truss subjected to a single load condition. This approximation approach is used to solve this multi-objective structural optimization model.

The remainder of this paper is organized in the following way. In Section 2, we have discussed about the structural optimization model. In Section 3, we have discussed about mathematics Prerequisites. In Section 4, we have discussed about extended n-ary t-norms. In Section 5, we have discussed about the calculation of some of special cases. In Section 6, we have discussed about weighted fuzzy aggregation. In Section 7, we have proposed the technique to solve multi-objective non-linear programming problem using extended t-norms based fuzzy optimization. In Section 8, we have solved multi-objective structural model using extended t-norms based fuzzy optimization. In Section 9, numerical solution of structural model of three bar truss and compared results by using different extended weighted t-norms. Finally, we draw conclusions in Section 10.

2 Multi-objective structural model

In the design of optimal structure i.e. lightest weight of the structure and minimum deflection of loaded joint that satisfies all stress constraints in members of the structure. To bar truss structure system the basic parameters (including the elastic modulus, material density, the maximum allowable stress, etc.) are known and the optimization’s target is that identify the optimal bar truss cross-section area so that the structure is of the smallest total weight, the minimum nodes displacement, in a given load conditions.

The multi-objective Structural model can be ex-pressed as: $\begin{matrix} minimize WT (A) \\ minimize δ (A) \\ subject to σ (A) \leq [σ] \\ A_{min} \leq A \leq A_{max} \end{matrix}$ (1)

where A = [A₁, A₂, …… , A_n] ^T are design variables for the cross section, n is the group number of design variables for the cross section bar, $WT (A) = \sum_{i = 1}^{n} ρ_{i} A_{i} L_{i}$ is the total weight of the structure, δ (A) is the deflection of loaded joint, L_i, A_i and ρ_i were the bar length, cross section area, and density of the ith group bars respectively. σ (A) is the stress constraint and [σ] is maximum allowable stress of the group bars under various conditions, A_min and A_max are the minimum and maximum cross section area respectively.

3 Prerequisite mathematics

3.1 Fuzzy set

Let X is a set (space), with a generic element of X denoted by x, that is X (x). Then a Fuzzy set (FS) is defined as $\tilde{A} = {(x, μ_{A} (x)) : x \in X}$ where $μ_{\tilde{A}} : X \to [0, 1]$ is the membership function of FS $\tilde{A}$ . $μ_{\tilde{A}} (x)$ is the degree of membership of the element x to the set $\tilde{A}$ .

3.2 α-Level set or α-cut of a fuzzy set

The α-level set of the fuzzy set $\tilde{A}$ of X is a crisp set A_α that contains all the elements of X that have membership values greater than or equal to α i.e. $\tilde{A} = {x : μ_{\tilde{A}} (x) \geq α, x \in X, α \in [0, 1]}$ .

3.3 Convex fuzzy set

A fuzzy set $\tilde{A}$ of the universe of discourse X is convex if and only if for all x₁, x₂ in X, $μ_{\tilde{A}} (λ x_{1} + (1 - λ x_{2}) \geq min (μ_{\tilde{A}} (x_{1}), μ_{\tilde{A}} (x_{2})))$

when 0 ≤ λ ≤ 1.

3.4 Normal fuzzy set

A fuzzy set $\tilde{A}$ of the universe of discourse X is called a normal fuzzy set implying that there exist at least one x ∈ X such that $μ_{\tilde{A}} (x) = 1$ .

3.5 Quasi t-norm

Let T : [0, 1] × [0, 1] → [0, 1] be a function satisfying the following axioms $\begin{matrix} a) T (a, b) = T (b, a), \forall a, b \in [0, 1] \\ b) T (T (a, b), c) = T (a, T (b, c)), \\ \forall a, b, c \in [0, 1] \\ c) T (a, b) \leq T (a, c) with b \leq c \forall a, b, c \in [0, 1] \\ d) T (0, 0) = 0, T (1, 1) = 1 \end{matrix}$

3.6 t-norm

A quasi-triangular norm T is called a triangular norm (or t-norm) if it satisfies T (a, 1) = a ; ∀ a ∈ [0, 1].

3.7 Extended n-ary quasi-t-norms

For the purpose of operations of multiple fuzzy sets, it is useful to define the notation of multi-dimensional t-norms. Let [0, 1] ⁿ be an n-dimensional cube and (x₁, x₂, x₃, …, x_n) , (z₁, z₂, …… , z_n) ∈ [0, 1] ⁿ.

A mapping T : [0, 1] ⁿ → [0, 1] is called ann-dimensional quasi-t-norm if it satisfies the following conditions:

$\begin{matrix} a) & T (x_{1}, \dots, x_{i - 1}, x_{i}, x_{i + 1}, \dots, x_{n}) = \\ T (x_{1}, \dots, x_{i - 1}, x_{j}, x_{i + 1}, \dots, x_{j - 1}, x_{i}, \\ x_{j + 1}, \dots, x_{n}) \\ b) & T (T (x_{1}, \dots, x_{n - 1}, x_{n}), x_{n + 1}, \dots, x_{2 n - 1}) \\ = T (x_{1}, \dots, x_{n - 1}, T (x_{n}, x_{n + 1}, \dots, x_{2 n - 1})) \\ c) & For (x_{1}, x_{2}, \dots, x_{n}) \leq (z_{1}, z_{2}, \dots, z_{n}) \Rightarrow \\ T (x_{1}, x_{2}, \dots, x_{n}) \leq T (z_{1}, z_{2}, \dots, z_{n}) with \\ for some i and x_{i} \leq z_{i} for some i, \\ i = 1, 2, 3, \dots, n . \\ d) & T (0, \dots, 0) = 0, T (1, \dots, 1) = 1 \end{matrix}$

4 Extended n-ary t-norms

A n-dimensional quasi-t-norm T is calledn-dimensional t-norm if it satisfies $T (1, 1, \dots, 1, x_{i}, 1, \dots, 1) = x_{i}$

Due to associative law it is easy to extend a triangular norm T into n arguments. The n-ary operation T_n on [0, 1] satisfies the following properties

(i) T_n (x₁, x₂, …, x_n) = T_n (x_σ1, x_σ2, …, x_σn) where σ is a permutation of {1, 2, … , n } (Commutativity) $\begin{matrix} (ii) T_{n} (x_{1}, x_{2}, \dots, x_{n}) \\ = T_{i + 1} (x_{1}, x_{2}, \dots, x_{i}, T_{n - i} (x_{i + 1}, \dots, x_{j}, \dots, x_{n})) \\ = T_{n - j + 1} (T (x_{1}, x_{2}, . . . ., x_{j}), x_{j + 1}, . . . . . ., x_{n}) \\ (Associativity) \end{matrix}$ $\begin{array}{l} (i N_{n}) (x i x_{i}^{'}) \\ T_{n} (x_{1}, x_{2}, ..., x_{n}) T_{n} (x_{1}^{'}, x_{2}^{'}, ..., x_{n}^{'}) \\ (Monotonocity) \end{array}$ $\begin{matrix} (iv) T_{n} (x_{1}, x_{2}, \dots, x_{i - 1}, 1, x_{i + 1}, \dots, x_{n}) \\ = T (x_{1}, x_{2}, \dots, x_{i - 1}, x_{i + 1}, \dots, x_{n}) \\ (Identity law) \end{matrix}$

A t-norm T_n is said to be continuous if T is continuous function on [0, 1]. From the above, we may call T_n an extension of triangular norm. In the sequel, we omit reference to the number of arguments n, and simply write T of the class of mapping generated by triangular norm T.

4.1 Four basic Binary t-norms and their generalization with weight factors

Minimum t-norm T_M (a, b) = min{ a, b } and extension in n-ary of this t-norm T_M (x₁, x₂, … , x_n) = min { x₁, x₂, …, x_n } and extended form with weights the above t-norm $T_{M}^{W} (x_{1}, W_{1}; x_{2}, W_{2}; \dots; x_{n}, W_{n}) = min {W_{1} x_{1}, W_{2} x_{2}, \dots, W_{n} x_{n}}$ Probabilistic t-norm T_P (a, b) = a . b and extension in n-ary of this t-norm $T_{P} (x_{1}, x_{2}, \dots, x_{n}) = x_{1} x_{2} \dots x_{n} = \prod_{i = 1}^{n} x_{i}$ and extended form with weights the above t-norm $T_{P}^{W} (x_{1}, W_{1}; x_{2}, W_{2}; \dots; x_{n}, W_{n}) = \prod_{i = 1}^{n} x_{i}^{W_{i}}$

Lukasiewicz t-norm (bounded t-norm) T_L (a, b) = max{ 0, a + b - 1 } and extension in n-ary of this t-norm $T_{L} (x_{1}, x_{2}, \dots, x_{n}) = max (\sum_{i = 1}^{n} x_{i} - (n - 1), 0)$ and extended form with weights the above t-norm $\begin{matrix} T_{L}^{W} (x_{1}, W_{1}; x_{2}, W_{2}; \dots; x_{n}, W_{n}) \\ = max (\sum_{i = 1}^{n} W_{i} x_{i} - (n - 1), 0) \end{matrix}$

Weber (or Drastic product) t-norm

$T_{D} (a, b) = {\begin{matrix} min (a, b) if max (a, b) = 1 \\ 0 otherwise and \end{matrix}$

extension in n-ary of this t-norm

$\begin{matrix} T_{D} (x_{1}, x_{2}, \dots, x_{n}) \\ = {\begin{matrix} min (x_{1}, x_{2}, \dots, x_{n}) if \\ max (x_{1}, x_{2}, \dots, x_{n}) = 1 \\ 0 otherwise \end{matrix} \end{matrix}$

and extended form with weights the above t-norm $\begin{matrix} T_{D}^{W} (x_{1}, W_{1}; x_{2}, W_{2}; \dots; x_{n}, W_{n}) \\ = {\begin{matrix} min (W_{1} x_{1}, W_{2} x_{2}, \dots, W_{n} x_{n}) if \\ max (W_{1} x_{1}, W_{2} x_{2}, \dots, W_{n} x_{n}) = 1 \\ 0 otherwise \end{matrix} \end{matrix}$

5 Some particular classes of t-norms and their special cases

A t-norm is commutative order semi-group with unit element 1 on [0, 1] of real numbers. So, the class of all t-norms is quite large. Some well-known classes of t-norm are discussed here.

I. Yager (1980) introduced the following class of t-norms $\begin{matrix} T_{λ}^{Y} (a, b) = 1 - min [1, {{(1 - a)}^{λ} + {(1 - b)}^{λ}}^{\frac{1}{λ}}] \\ \forall λ \in [0, \infty) \end{matrix}$

Extension in n-ary of the above t-norms $\begin{matrix} T_{λ}^{Y} (x_{1}, x_{2}, \dots \dots, x_{n}) \\ = 1 - min [1, {\sum_{i = 1}^{n} {(1 - x_{i})}^{λ}}^{\frac{1}{} λ}] \\ \forall λ \in [0, \infty) \end{matrix}$

The extended form with different weights of the above t-norms $\begin{matrix} T_{λ}^{WY} (x_{1}, W_{1}; x_{2}, W_{2}; \dots; x_{n} W_{2}) \\ = max [0, 1 -, {\sum_{i = 1}^{n} W_{i} {(1 - x_{i})}^{λ}}^{\frac{1}{λ}}] \\ W_{i} \geq 0, \forall λ \in [0, \infty) \end{matrix}$

So $\begin{matrix} (Ia) lim_{λ \to \infty} T_{λ}^{Y} (x_{1}, x_{2}, \dots \dots, x_{n}) = T_{M} \\ (x_{1}, x_{2}, \dots \dots, x_{n}) \\ (Ib) lim_{λ \to 0} T_{λ}^{Y} (x_{1}, x_{2}, \dots \dots, x_{n}) = \\ T_{D} (x_{1}, x_{2}, \dots \dots, x_{n}) \end{matrix}$

Proof of (Ia):

Case-I: If x₁ = x₂ = …… = x_i = …… = x_n ( ≠ 0) then from (a) we have $\begin{matrix} lim_{λ \to \infty} T_{λ}^{Y} (x_{1}, x_{2}, \dots \dots, x_{n}) \\ = lim_{λ \to \infty} [1 - min {1, n^{\frac{1}{λ}} (1 - x_{i})}] \\ = [1 - min {1, lim_{λ \to \infty} (n^{\frac{1}{λ}} (1 - x_{i}))}] \\ = [1 - min {1, (1 - x_{i})}] \\ = x_{i} = min {x_{1}, x_{2}, \dots \dots, x_{n}} = \\ T_{M} (x_{1}, x_{2}, \dots \dots, x_{n}) \end{matrix}$

Case II: If x₁ = x₂ = …… = x_i = …… =x_n = 0, then $\begin{matrix} lim_{λ \to \infty} T_{λ}^{Y} (x_{1}, x_{2}, \dots \dots, x_{n}) \\ = lim_{λ \to \infty} [1 - min {1, n^{\frac{1}{λ}}}] \\ = 0 = min {x_{1}, x_{2}, \dots \dots, x_{n}} \\ = T_{M} (x_{1}, x_{2}, \dots \dots, x_{n}) \end{matrix}$

Case III: If x_i ≠ x_j ∀ i, j = 1, 2, …, n and i ≠ j, then without loss of generality (due to commutativity), we assume x₁ < x₂ < …… … < x_i < …… < x_n. Let $P = {\sum_{i = 1}^{n} {(1 - x_{i})}^{λ}}^{\frac{1}{λ}}$ then using L’Hospital rule, we have $\begin{matrix} lim_{λ \to \infty} ln (P) = lim_{λ \to \infty} \frac{ln (\sum_{i = 1}^{n} {(1 - x_{i})}^{λ})}{λ} [\frac{\infty}{\infty} form] \\ = lim_{λ \to \infty} \frac{(\sum_{i = 1}^{n} {(1 - x_{i})}^{λ} ln (1 - x_{i}))}{\sum_{i = 1}^{n} {(1 - x_{i})}^{λ}} [\frac{\infty}{\infty} form] \\ = lim_{λ \to \infty} \frac{(ln (1 - x_{i}) + \sum_{i = 1}^{n - 1} {(\frac{1 - x_{i + 1}}{1 - x_{i}})}^{λ} ln (1 - x_{i + 1}))}{1 + \sum_{i = 1}^{n - 1} {(\frac{1 - x_{i + 1}}{1 - x_{i}})}^{λ}} \\ = ln (1 - x_{i}) \end{matrix}$

Hence $lim_{λ \to \infty} P = 1 - x_{i}$ and $\begin{matrix} lim_{λ \to \infty} T_{λ}^{Y} (x_{1}, x_{2}, \dots \dots, x_{n}) \\ = lim_{λ \to \infty} [1 - min {1, P}] \\ = x_{i} = T_{M} (x_{1}, x_{2}, \dots \dots, x_{n}) \end{matrix}$

Proof of (Ib):

Case-I: If x₁ = x₂ = … = x_i-1 = x_i+1 = … = x_n = 1 and x_i ≠ 1, we have $\begin{matrix} lim_{λ \to 0} T_{λ}^{Y} (x_{1}, x_{2}, \dots \dots, x_{n}) \\ = lim_{λ \to 0} [1 - min {1, (1 - x_{i})}] \\ = x_{i} = T_{D} (x_{1}, x_{2}, \dots \dots, x_{n}) \end{matrix}$

Case-II: If x_i (≠ 1) for i = 1, 2, …, n are alldistinct, $\begin{matrix} lim_{λ \to 0} T_{λ}^{Y} (x_{1}, x_{2}, \dots, x_{n}) = lim_{λ \to 0} [1 - min {1, n^{\frac{1}{λ}}}] \\ = 0 = T_{D} (x_{1}, x_{2}, \dots, x_{n}) \end{matrix}$

Case-III: If x_i = 1 for i = 1, 2, …, n, we have $\begin{matrix} lim_{λ \to 0} T_{λ}^{Y} (x_{1}, x_{2}, \dots \dots, x_{n}) = lim_{λ \to 0} [1 - min {1, 0}] \\ = 1 = T_{D} (x_{1}, x_{2}, \dots, x_{n}) \end{matrix}$

II. Hamacher (1978) introduced the following class of t-norms $\begin{matrix} T_{λ}^{H} (a, b) = \\ \frac{ab}{λ + (1 - λ) {1 - (1 - a) (1 - b)}}, \forall \in [0, \infty) \end{matrix}$

Extension in n-ary of the above t-norms $\begin{matrix} T_{λ}^{H} (x_{1}, x_{2}, \dots, x_{n}) \\ = \frac{\prod_{i = 1}^{n} x_{i}}{λ + (1 - λ) {1 - \prod_{i = 1}^{n} (1 - x_{i})}}, \forall \in [0, \infty) \end{matrix}$

The extended form with different weights of the above t-norms $\begin{matrix} T_{λ}^{WH} (x_{1}, W_{1}; x_{2}, W_{2}; \dots, x_{n}, W_{n}) \\ = \frac{\prod_{i = 1}^{n} (1 - W_{i} (1 - x_{i}))}{λ + (1 - λ) {\prod_{i = 1}^{n} W_{i} (1 - x_{i})}}, \\ \forall \in [0, \infty), W_{i} \geq 0 \end{matrix}$

(IIa) $lim_{λ \to 1} T_{λ}^{H} (x_{1}, x_{2}, \dots \dots, x_{n}) = T_{P} (x_{1}, x_{2}, \dots$ …, x_n)

(IIb) $lim_{λ \to \infty} T_{λ}^{H} (x_{1}, x_{2}, \dots \dots, x_{n}) = T_{D} (x_{1}, x_{2}, \dots$ …, x_n)

Proof of (IIa): $\begin{matrix} lim_{λ \to 1} T_{λ}^{H} (x_{1}, x_{2}, \dots, x_{n}) = \\ lim_{λ \to 1} \frac{\prod_{i = 1}^{n} x_{i}}{λ + (1 - λ) {1 - \prod_{i = 1}^{n} (1 - x_{i})}} \\ = \prod_{i = 1}^{n} x_{i} = T_{P} (x_{1}, x_{2}, \dots, x_{n}) \end{matrix}$

Proof of (IIb): $\begin{matrix} lim_{λ \to \infty} T_{λ}^{H} (x_{1}, x_{2}, \dots \dots, x_{n}) = \\ lim_{λ \to \infty} \frac{\prod_{i = 1}^{n} x_{i}}{λ + (1 - λ) {1 - \prod_{i = 1}^{n} (1 - x_{i})}} \end{matrix}$

Now if x₁ = x₂ = … = x_i-1 = x_i+1 = … = x_n = 1 then $lim_{λ \to \infty} T_{λ}^{H} (x_{1}, x_{2}, \dots \dots, x_{n}) = lim_{λ \to \infty}$ $\frac{x_{i}}{λ + (1 - λ) {1 - 0}} = x_{i}$ and if x₁≠ x₂ ≠ … ≠ x_i-1 ≠x_i+1 ≠ … ≠ x_n ≠ 1 then $lim_{λ \to \infty} T_{λ}^{H} (x_{1}, x_{2}, \dots \dots, x_{n}) = \frac{\prod_{i = 1}^{n} x_{i}}{\infty} = 0$

Thus $\begin{matrix} lim_{λ \to \infty} T_{λ}^{H} (x_{1}, x_{2}, \dots, x_{n}) \\ = {\begin{matrix} x_{j} if x_{j} = 1 (j = 1, 2, \dots, i - 1, i + 1, \dots, n) \\ 0 otherwise \end{matrix} \\ = T_{D} (x_{1}, x_{2}, \dots, x_{n}) \end{matrix}$

III. Dombi (1982) introduced the following class of t-norms $\begin{matrix} T_{λ}^{Dom} (a, b) = \\ \frac{ab}{1 + {{(\frac{1}{a} - 1)}^{λ} + {(\frac{1}{b} - 1)}^{λ}}^{\frac{1}{λ}}}, \forall λ \in [0, \infty) \end{matrix}$

Extension in n-ary of the above t-norms $\begin{matrix} T_{λ}^{Dom} (x_{1}, x_{2}, \dots, x_{n}) = \\ \frac{1}{1 + {\sum_{i = 1}^{n} {(\frac{1}{x_{i}} - 1)}^{λ}}^{\frac{1}{λ}}}, \forall \in [0, \infty) \end{matrix}$

The extended form with different weights of the above t-norms $\begin{matrix} T_{λ}^{WDom} (x_{1}, W_{1}; x_{2}, W_{2}; \dots, x_{n}, W_{n}) = \\ \frac{1}{1 + {\sum_{i = 1}^{n} W_{i} {(\frac{1}{x_{i}} - 1)}^{λ}}^{\frac{1}{λ}}}, \forall λ \in [0, \infty), W_{i} \geq 0 \end{matrix}$

Here the following limit holds well, $\begin{matrix} (IIIa) lim_{λ \to \infty} T_{λ}^{Dom} (x_{1}, x_{2}, \dots \dots, x_{n}) \\ = T_{M} (x_{1}, x_{2}, \dots \dots, x_{n}) \end{matrix}$ $\begin{matrix} (IIIa) lim_{λ \to 1} T_{λ}^{Dom} (x_{1}, x_{2}, \dots \dots, x_{n}) \\ = T_{D} (x_{1}, x_{2}, \dots \dots, x_{n}) \end{matrix}$

Proof of (IIIa) and (IIIb) same as (Ia) and (Ib).

6 Weighted fuzzy aggregation

Weighted Aggregation has been used quite extensively especially in fuzzy decision-making, where the weights are used to represent the relative importance the decision maker attaches to different decision criterion (goals or constraints). Weighted aggregation of fuzzy sets by using t-norms has been considered first by Yager [13]. He proposed to modify the membership functions with the associated weight factors before the fuzzy aggregation. The weighted aggregation is then the aggregation of the modified membership functions.

A general form of this idea gives the weighted aggregation function [14] $D (x, W) = T [I (μ_{1} (x), W_{1}), . . ., I (μ_{k} (x), W_{k})]$ (2)

where W is a vector of weight factor W_i ∈ [0, 1] i = 1, 2, …, k associated with the aggregated membership function μ_i (x). T is t-norm and I is a function of two variables that transforms the membership functions with $\sum_{i = 1}^{k} W_{i} = 1, W_{i} \geq 0$ .

7 Mathematical analysis

7.1 General Fuzzy Non-linear Programming (FNLP) Technique to solve Multi-bjective Non-Linear Programming Problem (MONLP)

A Multi-Objective Non-Linear Programming (MONPL) or Vector Minimization problem (VMP) may be taken in the following form: $Min f (x) = [f_{1} (x), f_{2} (x), \dots \dots \dots f_{k} (x)]^{T}$ (3)

Subject to

$x \in X = {\begin{matrix} x \in R^{n} : g_{j} (x) \leq or = or \geq b_{j} \\ for j = 1, 2, 3, \dots \dots, m \end{matrix}} and$

l_i ≤ x_i ≤ u_i (i = 1, 2, 3, …, n).

Zimmermann [28] showed that fuzzy programming technique can be used to solve the multi-objective programming problem.

To solve MONLP problem, following steps are used:

Step 1: Solve the MONLP (3) as a single objective non-linear programming problem using only one objective at a time and ignoring the others, these solutions are known as ideal solution.

Step 2: From the result of step 1, determine the corresponding values for every objective at each solution derived. With the values of all objectives at each ideal solution, pay-off matrix can be formulated asfollows:

$\begin{matrix} f_{1} (x) f_{2} (x) \dots f_{k} (x) \\ \begin{matrix} x^{1} \\ x^{2} \\ \dots \\ x^{k} \end{matrix} (\begin{matrix} f_{1}^{*} (x^{1}) f_{2} (x^{1}) \dots f_{k} (x^{1}) \\ f_{1} (x^{2}) f_{2}^{*} (x^{2}) \dots f_{k} (x^{2}) \\ \dots \dots \dots \dots \\ f_{1} (x^{k}) f_{2} (x^{k}) \dots f_{k}^{*} (x^{k}) \end{matrix}) \end{matrix}$

Here x¹ x², x³, …, x^k are the ideal solutions of the objectives f₁ (x) , f₂ (x) , …, f_k (x), respectively. The maximum value of each column U_r gives upper bound or upper tolerance or highest acceptable level of achievement for the rth objective function f_r (x), where U_r = max{ f_r (x¹) , f_r (x²) , …… , f_r (x^k) } and the minimum value of each column L_r gives lower bound or lower tolerance limit or aspired level of achievement for the rth objective function f_r (x) where L_r = min{ f_r (x¹) , f_r (x²) , …… , f_r (x^k) } for r = 1, 2, …, k.

Step 3: Using aspiration level of each objective of the MONLP (3) may be written as follows:

Find x so as to satisfy $\begin{matrix} f_{r} (x) \tilde{\leq} L_{r} with tolerance P_{r} (= U_{r} - L_{r}) \\ for r = 1, 2, 3, \dots \dots, k \end{matrix}$ $x \in X . l_{i} \leq x_{i} \leq u_{i} (i = 1, 2, 3, \dots, n)$

Here objective functions of (3) are considered as fuzzy constraints. These types of fuzzy constraints can be quantified by eliciting a corresponding membership function: as (4) and with Fig. 1. $\begin{matrix} μ_{r} (f_{r} (x)) = {\begin{matrix} 0 if f_{r} (x) \geq U_{r} \\ \frac{U_{r} - f_{r} (x)}{U_{r} - L_{r}} if L_{r} \leq f_{r} (x) \leq U_{r} \\ 1 if f_{r} (x) \leq L_{r} \end{matrix} \\ for r = 1, 2, 3, \dots, k \end{matrix}$ (4)

After determining the different membership functions for each of the objective functions, one can adopt following two types of fuzzy decision are

Yager t-norm

Hamacher t-norm

Dombi t-norm

(i) According to the extension of the weighted Yager t-norm operator

$\begin{matrix} Maximize μ_{\tilde{D}}^{Y} (x; W) \\ = Maximize (0, 1 - \sqrt[λ]{\sum_{r = 1}^{k} W_{r} (1 - μ_{r} {(f_{r} (x))}^{λ}}), \\ λ > 0 \\ subject to 0 \leq μ_{r} (f_{r} (x)) \leq 1, \\ x \in X, W_{r} \geq 0, \sum_{r = 1}^{k} W_{r} = 1; \end{matrix}$ (5)

(ii) According to the extension of the weighted Hamacher t-norm operator $\begin{matrix} Maximize μ_{\tilde{D}}^{H} (x; W) \\ = \frac{1}{λ + (1 - λ) \sum_{r = 1}^{k} W_{r} [\frac{(1 - μ_{r} (f_{r} (x)))}{μ_{r} (f_{r} (x))}]} λ \leq 2 \end{matrix}$ (6)

subject to the same constraints of (5)

(iii) According to the extension of the weighted Dombi t-norm operator $\begin{matrix} Maximize μ_{\tilde{D}}^{Dom} (x; W) \\ = Maximize [\frac{1}{1 + {\sum_{i = 1}^{n} W_{i} {(\frac{1}{μ_{r} (f_{r} (x))} - 1)}^{λ}}^{\frac{1}{λ}}}], \\ λ > 0 \end{matrix}$ (7)

subject to the same constraints of (5)

Step 4: Solving any one among five Equations (5 to 7) we will get optimal solution of (3).

7.2 Complete optimal solution

x^* is said to be a complete optimal solution to the MONLP (3) if and only if there exists x ∈ X such that f_r (x^*) ≤ f_r (x) for r = 1, 2, …, k and for all x ∈ X. However, when the objective functions of the MONLP conflict with each other, a complete optimal solution does not always exist and hence the Pareto Optimality Concept arises and it is defined as follows.

7.3 Pareto optimal solution

x^* is said to be a Pareto optimal solution to the MONLP (3) if and only if there does not exist another x ∈ X such that f_r (x^*) ≤ f_r (x) for all r = 1, 2, …, k and f_j (x) ≠ f_j (x^*) for at least one j j∈ { 1, 2, …, k }.

8 Fuzzy programming technique in Multi-Objective Structural Model

To solve the above MOSOP (1), step 1 of (7.1) is used. After that according to step 2 pay-off matrix formulated as follows: $\begin{matrix} WT (A) δ (A) \\ \begin{matrix} A^{1} \\ A^{2} \end{matrix} (\begin{matrix} {WT}^{*} (A^{1 *}) δ (A^{1 *}) \\ {WT}^{*} (A^{2 *}) δ (A^{2 *}) \end{matrix}) \end{matrix}$

After that according to step 2 of 7.1, the bounds are U₁ = max{ WT (A^1*) , WT (A^2*) }, L₁ = min{ WT (A^1*) , WT (A^2*) } for weight function WT (A) (where L₁ ≤ WT (A) ≤ U₁) and the bounds of objective are U₂ = max{ δ (A^1*) , δ (A^2*) }, L₂ = min{ δ (A^1*) , δ (A^2*) } for deflection function δ (A) (where L₂ ≤ δ (A) ≤ U₂) are identified.

Above MOSOPP reduces to a FMOSOPP as follows;

Find A

such that

$WT (A) \tilde{\leq} L_{1}$ with maximum allowable tolerance P₁ (= U₁ - L₁)

$δ (A) \tilde{\leq} L_{2}$ with maximum allowable tolerance P₂ (= U₂ - L₂) $\begin{matrix} σ (A) \leq [σ_{0}] \\ A_{min} \leq A \leq A_{max} \end{matrix}$

Here for simplicity linear membership functions $μ_{WT} (WT (A))$ and $μ_{δ} (δ (A))$ for the objective functions WT (A) and δ (A) respectively are defined as follows: $\begin{matrix} μ_{WT} (WT (A)) \\ = {\begin{matrix} 1 if WT (A) \leq L_{1} \\ (\frac{U_{1} - WT (A)}{U_{1} - L_{1}}) if L_{1} \leq WT (A) \leq U_{1} \\ 0 if WT (A) \geq U_{1} \end{matrix} \end{matrix}$ $μ_{δ} (δ (A)) = {\begin{matrix} 1 if δ (A) \leq L_{2} \\ (\frac{U_{2} - δ (A)}{U_{2} - L_{2}^{'}}) if L_{2} \leq δ (A) \leq U_{2} \\ 0 if δ (A) \geq U_{2} \end{matrix}$

After determining the different membership functions for each of the objective functions, one can adopt following types fuzzy decision using t-norms are

i) According to the extension of the weighted Yager t-norm operator with λ = 2 $\begin{matrix} maximize \\ (0, 1 - \sqrt{W_{1} {(1 - μ_{WT} (WT (A)))}^{2} + W_{2} {(1 - μ_{δ} (δ (A)))}^{2}}) \end{matrix}$

$\begin{matrix} subject to \\ 0 \leq μ_{WT} (WT (A)) \leq 1, \\ 0 \leq μ_{δ} (δ (A)) \leq 1, \\ σ (A) \leq [σ], \\ A_{min} \leq A \leq A_{max}, \\ W_{1} \geq 0, W_{2} \geq 0, W_{1} + W_{2} = 1; \end{matrix}$ (8)

ii) According to the extension of the weighted Hamacher t-norm operator λ = 0

$\begin{matrix} Maximize μ_{\tilde{D}}^{H} (x; W) \\ = \frac{1}{1 + \sum_{r = 1}^{k} W_{r} [\frac{(1 - μ_{r} (f_{r} (x)))}{μ_{r} (f_{r} (x))}]} \end{matrix}$ (9)

subject to the same constraints of (8)

iii) According to the extension of the weighted Dombi t-norm operator with λ = 2

$\begin{matrix} Max μ_{\tilde{D}}^{Dom} (x; W) \\ = Max [\frac{1}{1 + {W_{1} {(\frac{1}{μ_{WT} (WT (A))} - 1)}^{2} + W_{2} {(\frac{1}{μ_{δ} (δ (A))} - 1)}^{2}}^{\frac{1}{2}}}] \end{matrix}$ (10)

subject to the same constraints of (8)

Solving any one among four equations (8) to (10) we will get optimal solution of (1).

9 Numerical solution of a multi-objective structural optimization model of a three-bar truss

A well-known three bar [21] planar truss structure is considered as Fig. 2. The design objective is to minimize weight of the structural WT (A₁, A₂) and minimize the deflection δ (A₁, A₂) along u and v at loading point of a statistically loaded planar truss subjected to stress σ_i (A₁, A₂) constraints on each of the truss members i = 1, 2, 3.

The multi-objective optimization problem can be stated as follows:

$\begin{matrix} Minimize WT (A_{1}, A_{2}) & = & ρ L (2 \sqrt{2} A_{1} + A_{2}) \\ minimize δ_{u} (A_{1}, A_{2}) & = & \frac{\sqrt{2} PL}{{EA}_{1}} \\ minimize δ_{v} (A_{1}, A_{2}) & = & \frac{\sqrt{2} PL}{E (A_{1} + \sqrt{2} A_{2})} \end{matrix}$ (11)

$\begin{matrix} subject to σ_{1} (A_{1}, A_{2}) & \equiv & \frac{P (2 A_{1} + A_{2})}{2 A_{1} A_{2} + 2 A_{1}^{2}} \leq [σ_{1}^{T}] \\ σ_{2} (A_{1}, A_{2}) & \equiv & \frac{P}{(A_{1} + \sqrt{2} A_{2})} \leq [σ_{2}^{T}] \\ σ_{3} (A_{1}, A_{2}) & \equiv & \frac{{PA}_{2}}{2 A_{1} A_{2} + 2 A_{1}^{2}} \leq [σ_{3}^{C}] \\ A_{i}^{min} \leq A_{i} \leq A_{i}^{max}; i = 1, 2 \end{matrix}$

The input data for MOSOP(11) is given as follows:

P (applied load) = 20 KN, ρ(material density) = 100KN/m³, L (length of each bar) = 1m, $[σ_{1}^{T}]$ (maximum allowable tensile stress for bar 1) = 20 KN/m², $[σ_{2}^{T}]$ (maximum allowable tensile stress for bar 2) = 20 KN/m², $[σ_{3}^{C}]$ (maximum allowable compressive stress for bar 3) = 15 KN/m², E(Young’s modulus) =2 × 10⁸KN/m², 0.1 × 10^-4m² ≤ A₁, A₂ ≤ 5 ×10^-4m²

Solution: According to step 2 pay off matrix is formulated as follows: $\begin{matrix} WT (A_{1}, A_{2}) δ_{u} (A_{1}, A_{2}) δ_{v} (A_{1}, A_{2}) \\ \begin{matrix} A^{1} \\ A^{2} \\ A^{3} \end{matrix} (\begin{matrix} 3 . 732051 1 . 267949 0 . 7320508 \\ 15 . 37659 0 . 2828427 0 . 2096440 \\ 19 . 14214 0 . 2828427 0 . 1171573 \end{matrix}) \end{matrix}$

Here U₁ = 19.14214, L₁ = 3.732051, U₂ = 1 . 267949, L₂ = 0.2828427, U₃ = 0.7320508, L₃ = 0.1171573. Here linear membership function for the objective functions WT (A₁, A₂), δ_u (A₁, A₂) and δ_v (A₁, A₂) are defined as follow $\begin{matrix} μ_{WT} (WT (A_{1}, A_{2})) \\ = {\begin{matrix} 1 if WT (A_{1}, A_{2}) \leq 3.732051 \\ \frac{19.14214 - WT (A_{1}, A_{2})}{15.410089} \\ if 3.732051 \leq WT (A_{1}, A_{2}) \leq 19.14214 \\ 0 if WT (A_{1}, A_{2}) \geq 19.14214 \end{matrix} \end{matrix}$ $\begin{matrix} μ_{δ} (δ_{u} (A_{1}, A_{2})) \\ = {\begin{matrix} 1 if δ_{u} (A_{1}, A_{2}) \leq 0.2828427 \\ \frac{1 . 267949 - δ_{u} (A_{1}, A_{2})}{0.9851063} \\ if 0.2828427 \leq δ_{u} (A_{1}, A_{2}) \leq 1.267949 \\ 0 if δ_{u} (A_{1}, A_{2}) \geq 1.267949 \end{matrix} \end{matrix}$ $\begin{matrix} μ_{δ} (δ_{v} (A_{1}, A_{2})) \\ = {\begin{matrix} 1 if δ_{v} (A_{1}, A_{2}) \leq 0.1171573 \\ \frac{0.7320508 - δ_{v} (A_{1}, A_{2})}{0.6148935} \\ if 0.1171573 \leq δ_{v} (A_{1}, A_{2}) \leq 0.7320508 \\ 0 if δ_{v} (A_{1}, A_{2}) \geq 14 . 64102 \end{matrix} \end{matrix}$

The optimal results of model (11) using different t-norms and weights are shown in Tables 1 to 4.

For equal importance, the extension of the weighted parameterized Dombi t-norm operator gives minimum structural weight where as the extension of the weighted parameterized Hamacher t-norm operator gives minimum deflection.

For more importance on Structural Weight, the extension of the weighted parameterized Hamacher t-norm operator gives minimum weight.

For more importance on deflection, the extension of the weighted parameterized Hamacher t-norm operator gives minimum deflection along u.

For more importance on deflection, the extension of the weighted parameterized Hamacher t-norm operator gives minimum deflection along v.

10 Conclusions

In this paper, we have proposed a method to solve multi-objective structural model. Here binary t-norms are expressed in extended n-ary t-norms and discussed their basic properties and some special cases. The said model is converted into an equivalent single objective problem and it is solved by using t-norms based fuzzy decision making technique. A main advantage of the proposed method is that it allows the user to concentrate on the actual limitations in a problem during the specification of the flexible objectives. This approximation method can be applied to optimize different models in various fields of engineering and sciences.

Footnotes

Acknowledgments

The authors would like to thank anonymous referees for review of this paper.

References

Alsina

, Trillas

and Valverde

, On some logical connectives for fuzzy set theory, J Math Anal Appl 93 (1981), 15–26.

Dede

, Bekiroglu

and Ayvaz

, Weight minimization of trusses with genetic algorithm, Appl Soft Comput 11(2) (2011), 2565–2575.

Dey

Samir

and Roy

Tapan Kumar

, A fuzzy programming technique for solving multi-objective structural problem, International Journal of Engineering and Manufacturing 5 (2014), 24–42.

Gupta

M.M.

and Qi

, theory of t-norms and fuzzy inference methods, Fuzzy sets and systems 40 (1991), 431–450.

Eschenauer

Hans

, Koski

Juhani

and Osyczka

Andrzej

, editors. Multicriteria Design Optimization. Procedures and Applications. Springer-Verlag, 1990.

Kaveh

and Talatahari

, An improved ant colony optimization for the design of planar steel frames, Eng Struct 32(3) (2010), 864–873.

Kaveh

and Khayatazad

, Ray optimization for sizeand shape optimization of truss structures, Computers and Structures 117 (2013), 82–94.

Kaymak

and Sousa

J.M.

, Weighted constraints in fuzzy optimization, ERS–200119-LIS.

Luh

G.C.

and Lin

C.Y.

, Optimal design of truss-structures using particle swarm optimization, Computers and Structures 89(2324) (2011), 2221–2232.

10.

Pei

, Pan

, Xu

and Quin

, on some properties of fuzzy T^*-algebras and fuzzy F^*-algebras, International Journal of Fuzzy Mathematics 11 (2003), 7–16.

11.

Perez

R.E.

and Behdinan

, Particle swarm approach for structural design optimization, Computers & Structures 85(19-20) (2007), 1579–1588.

12.

Bellman

R.E.

and Zadeh

L.A.

, Decision-making in a fuzzy environment, Management Science 17(4) (1970), 141–164.

13.

Yager

R.R.

, Fuzzy decision making including unequal objectives, Fuzzy Sets and Systems 1 (1978), 87–95.

14.

Yager

R.R.

, General multiple-objective decision functions and linguistically quantified statements, International Journal of Man-Machine Studies 21 (1984), 389–400.

15.

Rao

, Game theory approach for multi-objective structural optimization, Computers and Structures 25(1) (1986), 119–127.

16.

Rao

S.S.

, Multi-objective optimization in structural design with uncertain parameters and stochastic processes, AIAA Journal 22(11) (1984), 1670–1678.

17.

Rao

S.S.

, Multi-objective optimization of fuzzy structural systems, International Journal for Numerical Methods in Engineering 24(1) (1987), 1157–1171.

18.

Rao

S.S.

, Description and optimum design of fuzzy mathematical systems, Journal of Mechanisms, Transmissions, and Automation in Design 109 (1987), 126–132.

19.

Ruan

and Kerre

E.E.

, Fuzzy implication operators and generalized fuzzy method of cases, Fuzzy Sets and Systems 54 (1993), 23–37.

20.

Samanta

and Roy

T.K.

, Extended t-norm and its application in portfolio selection model, Journal of Fuzzy Mathematics, USA 13(4) (2005), 995–1010.

21.

Shih

C.J.

, Chi

C.C.

and Hsiao

J.H.

, Alternative α-level-cuts methods for optimum structural design with fuzzy resources, Computers and Structures 81 (2003), 2579–2587.

22.

Shih

C.J.

and Lee

H.W.

, Level-cut approaches of first and second kind for unique solution design in fuzzy engineering optimization problems, Tamkang Journal of Science and Engineering 7(3) (2004), 189–198.

23.

Sonmez

, Discrete optimum design of truss structures using artificial bee colony algorithm, Struct Multidiscip Optimiz 43(1) (2011), 85–97.

24.

Wang

G.Y.

and Wang

W.Q.

, Fuzzy optimum design of structure, Engineering Optimization 8 (1985), 291–300.

25.

, Fuzzy optimization of structures by the two-phase method, Computer and Structure 31(4) (1989), 575–580.

26.

Yeh

Y.C.

and Hsu

D.S.

, Structural optimization with fuzzy parameters, Computer and Structure 37(6) (1990), 917–24.

27.

Zadeh

L.A.

, Fuzzy set, Information and Control 8(3) (1965), 338–353.

28.

Zimmermann

H.J.

, Fuzzy linear programming with several objective function, Fuzzy Sets and Systems 1 (1978), 46–55.