Neutrosohic Sets and Systems Vol 14 016 93 University of New Mexico Otimal Design of Truss Structures Using a Neutrosohic Number Otimization Model under an Indeterminate Environment Wenzhong Jiang & Jun Ye Deartment of Electrical and Information Engineering and Deartment of Civil Engineering Shaoxing University 508 Huancheng West Road Shaoxing Zhejiang Province 31000 PR China Corresonding author: Jun Ye E-mail: yehjun@aliyuncom Abstract This aer defines basic oerations of neutrosohic numbers and neutrosohic number functions for objective functions and constraints in otimization models Then we roose a general neutrosohic number otimization model for the otimal design of truss structures The alication and effectiveness of the neutrosohic number otimization method are demonstrated through the design examle of a two-bar truss structure under indeterminate environment to achieve the minimum weight objective under stress and stability constraints The comarison of the neutrosohic number otimal design method with traditional otimal design methods roves the usability and suitability of the resented neutrosohic number otimization design method under an indeterminate/neutrosohic number environment Keywords: Neutrosohic number neutrosohic number function neutrosohic number otimization model neutrosohic number otimal solution truss structure design 1 Introduction In the real-world there is incomlete unknown and indeterminate information How to exress incomlete unknown and indeterminate information is an imortant roblem Hence Smarandache [1-3] firstly introduced a concet of indeterminacy which is denoted by the symbol I as the imaginary value and defined a neutrosohic number as N = a + bi for a b R (all real numbers which consists of both the determinate art a and the indeter-minate art bi So it can exress determinate and/or inde-terminate information in incomlete uncertain and inde-terminate roblems After that Ye [4 5] alied neutro-sohic numbers to decision making roblems Then Kong et al [6] and Ye [7] alied neutrosohic numbers to fault diagnosis roblems under indeterminate environments Further Smarandache [8] introduced an interval function (so-called neutrosohic function/thick function g(x = [g 1 (x g (x] for x R to describe indeterminate roblems by the interval functions And also Ye et al [9] introduced neutrosohic/interval functions of the joint roughness coef-ficient and the shear strength in rock mechanics under in-determinate environments It is obvious that neutrosohic numbers are very suitable for the exression of determinate and/or indeterminate information Unfortunately existing otimization design methods [10-13] cannot exress and deal with indeterminate otimization design roblems of engineering structures under neutrosohic number environments Furthermore the Smarandache s neutrosohic function [8] cannot also exress such an indeterminate function involving neutrosohic numbers Till now there are no concets of neutrosohic number functions and neutrosohic number otimization designs in all existing literature Therefore one has to define new functions containing NNs to handle indeterminate otimization roblems of engineering designs under a neutrosohic number environment To handle this issue this aer firstly defines a new concet of neutrosohic number functions for the neutrosohic number objective functions and constraints in engineering otimization design roblems with determinate and indeterminate information and then rooses a general neutrosohic number otimization model and a solution method to realize neutrosohic number otimization roblems of truss structure design where the obtained neutrosohic number otimal solution can satisfy the design requirements in indeterminate situations The remainder of this aer is structured as follows Section defines some new concets of neutrosohic number functions to establish the neutrosohic number objective functions and constraints in indeterminate otimization design roblems and rooses a general neutrosohic number otimization model for truss structure designs In Section 3 the neutrosohic number otimal design of a two-bar truss structure is resented under a neutrosohic number environment to illustrate the alication and effectiveness of the roosed neutrosohic number otimization design method Section 4 contains some conclusions and future research directions Wenzhong Jiang Jun Ye Otimal Design of Truss Structures Using a Neutrosohic Number Otimization Model under an Indeterminate Environment
94 Neutrosohic Sets and Systems Vol 14 016 Neutrosohic numbers and otimization models 1 Some basic oerations of neutrosohic numbers It is well known that there are some indeterminate design arameters and alied forces in engineering design roblems For examle the allowable comressive stress of some metal material is given in design handbooks by a ossible range between 40 MPa and 460 MPa denoted by = [40 460] which reveals the value of is an indeterminate range within the interval [40 460] Then a neutrosohic number N = a + bi for a b R (all real numbers can effectively exress the determinate and/or indeterminate information as N = 40 + 40I for I [0 1] where its determinate art is a = 40 its indeterminate art bi = 40I and the symbol I denotes indeterminacy and belongs to the indeterminate interval [inf I su I] = [0 1] For another examle if some external force is within [000 500] kn then it can be exressed as the neutrosohic number N = 000 + 50I kn for I [0 10] or N = 000 + 5I kn for I [0 100] corresonding to some actual requirement It is noteworthy that there are N = a for bi = 0 and N = bi for a = 0 in two secial cases Clearly the neutrosohic number can easily exress its determinate and/or indeterminate information where I is usually secified as a ossible interval range [inf I su I] in actual alications Therefore neutrosohic numbers can easily and effectively exress determinate and/or indeterminate information under indeterminate environments For convenience let Z be all neutrosohic numbers (Z domain then a neutrosohic number is denoted by N = a + bi = [a + b(inf I a + b(su I] for I [inf I su I] and N Z For any two neutrosohic numbers N 1 N Z we can define the following oerations: (1 ( N N a a ( b b I 1 1 1 [ a a b (inf I b (inf I 1 1 a a b (su I b (su I] 1 1 N N a a ( b b I 1 1 1 [ a a b (inf I b (inf I 1 1 a a b (su I b (su I] 1 1 ; ; (3 (4 N N a a ( a b a b I b b I 1 1 1 1 1 ( a1 b1 (inf I( a b (inf I ( 1 1(inf ( (su min a b I a b I ( a1 b1 (su I( a b (inf I ; ( a1 b1 (su I( a b (su I ( a1 b1 (inf I( a b (inf I ( a1 b1 (inf I( a b (su I max ( a1 b1 (su I( a b (inf I ( a1 b1 (su I( a b (su I N1 a1 b1 I [ a1 b1 (inf I a1 b1 (su I] N a b I [ a b (inf I a b (su I] a1 b1 (inf I a1 b1 (inf I a b (su I a b (inf I min a1 b1 (su I a1 b1 (su I a b (su I a b (inf I a1 b1(inf I a1 b1(inf I a b(su I a b(inf I max a1 b1 (su I a1 b1 (su I a b (su I a b (inf I Neutrosohic number functions and neutrosohic number otimization model In engineering otimal design roblems a general otimization model consists of the objective function and constrained functions In indeterminate otimization roblems of engineering designs then objective functions and constrained functions may contain indeterminate information To establish an indeterminate otimization model in a neutrosohic number environment we need to define neutrosohic number functions in Z domain Definition 1 A neutrosohic number function with n design variables in Z domain is defined as F(X I: Z n Z (1 where X = [x 1 x x n ] T for X Z n is a n-dimensional vector and F(X I is either a neutrosohic number linear function or a neutrosohic number nonlinear function For examle F1 ( X I (1 I x1 x (3 I for X = [x 1 x ] T Z is a neutrosohic number linear function then F ( X I Ix1 (3 I x for X = [x 1 x ] T Z is a neutrosohic number nonlinear function Wenzhong Jiang Jun Ye Otimal Design of Truss Structures Using a Neutrosohic Number Otimization Model under
Neutrosohic Sets and Systems Vol 14 016 95 3 General neutrosohic number otimization model Generally seaking neutrosohic number otimization design roblems with n design variables in Z domain can be defined as the general form of a neutrosohic number otimization model: min F(X I st G k (X I 0 k = 1 m ( H j (X I=0 j = 1 s XZ n I [inf I su I] where F(X I is a neutrosohic number objective function and G 1 (x G (x G m (x and H 1 (x H (x H s (x: Z n Z are neutrosohic number inequality constraints and neutrosohic number equality constraints resectively for X Z n and I [inf I su I] However if the neutrosohic number otimal solution of design variables satisfies all these constrained conditions in a neutrosohic number otimization model the otimal solution is feasible and otherwise is unfeasible Generally seaking the otimal solution of design variables and the value of the neutrosohic number objective function usually are neutrosohic numbers/interval ranges (but not always To solve the neutrosohic number otimization model ( we use the Lagrangian multiliers for the neutrosohic number otimization model Then the Lagrangian function that one minimizes is structured as the following form: L( X μ λ F( X I m s (3 G ( X I H ( X I k k j j k1 j1 Z m Z s XZ n I[inf I su I] The common Karush-Kuhn-Tucker (KKT necessary conditions are introduced as follows: m F( X I { G ( X I} { H ( X I} 0 (4 k k j j k1 j1 s 3 Otimal design of a two-bar truss structure under a neutrosohic number environment To demonstrate the neutrosohic number otimal design of a truss structure in an indeterminate environment a simly two-bar truss structure is considered as an illustrative design examle and showed in Fig1 In this examle the two bars use two steel tubes with the length L in which the wall thick is T=5mm The otimal design is erformed in a vertically external loading case The vertical alied force is F = (3+04I10 5 N the material Young s modulus and density E=110 5 MPa and ρ = 7800 kg/m 3 resectively and the allowable comressive stress is = 40 + 40I The otimal design objective of the truss structure is to minimize the weight of the truss structure in satisfying the constraints of stress and stability In this class of otimization roblems the average diameter D of the tube and the truss height H are taken into account as two design variables denoted by the design vector X = [x 1 x ] T = [D H] Due to the geometric structure symmetry of the twobar truss we only consider the otimal model of one bar of both First the total weight of the tube is exressed by the following formula: 1/ M AL Tx 1(B x where A is the cross-sectional area A = Tx 1 and B is the distance between two suorting oints Then the comressive force of the steel tube is 1/ FL F( B x F1 x x where L is the length of the tube and F 1 is the comressive force of the tube Thus the comressive stress of the tube is reresented as the following form: 1/ F1 F( B x A Tx1x Hence the constrained condition of the strength for the tube is written as 1/ F( B x Tx x 1 combined with the original constraints comlementary slackness for the inequality constraints and k 0 for k = 1 m However it may be difficult to solve neutrosohic nonlinear otimization models in indeterminate nonlinear otimization design roblems such as multile-bar truss structure designs under neutrosohic number environments by the Karush-Kuhn-Tucker (KKT necessary conditions Hence this aer will research on the neutrosohic number otimization design roblem of a simle two-bar truss structure in the following section to realize the rimal investigation of the truss structure otimal design in a neutrosohic number environment Fig 1 Two-bar truss structure Wenzhong Jiang Jun Ye Otimal Design of Truss Structures Using a Neutrosohic Number Otimization Model under
96 Neutrosohic Sets and Systems Vol 14 016 For the stability of the comressive bar the critical force of the tube is given as follows: EW EA( T x F c I 1 L 8( B x where W I is the inertia moment of the cross-section of the tube The critical stress of the tube is given as Fc E( T x c A B x 1 8( Thus the constrained condition of the stability for the tube is written as F( B x E( T x 1/ 1 Tx1 x 8( B x Finally the neutrosohic otimization model of the truss structure can be formulated as: min M( X I Tx ( B x 1/ 1 1/ F( B x s t G1 ( X I 0 Tx1x F( B x E( T x G ( 0 1/ 1 X I Tx1x 8( B x By solving the neutrosohic otimization model the neutrosohic number otimal solution of the two design variables is given as follows: F x 1 X T(40 40 I x B 5 1414(15 0 I 10 785(40 40 I 760 In this case the neutrosohic number otimal value of the objective function is obtained as follows: 4FB 371(15 0 I M ( X I (40 40 I Since there exists the indeterminacy I in these neutrosohic number otimal values it is necessary that we discuss them when the indeterminacy I is secified as ossible ranges according to actual indeterminate requirements in the actual alication Obviously the neutrosohic number otimization roblem reveals indeterminate otimal results (usually neutrosohic number otimal solutions but not always If the indeterminacy I is secified as different ossible ranges of I =0 I [0 1] I [1 3] I [3 5] I [5 7] and I [7 10] for convenient analyses then all the results are shown in Table 1 Table 1 Otimal results of two-bar truss structure design in different secified ranges of I [inf I su I] I [inf I su I] D =x 1 (mm H =x (mm M(X I (kg I = 0 64331 760 84686 I [0 1] [58737 79087] 760 [773 95977] I [1 3] [567068 831] 760 [74649 10850] I [3 5] [610109 83393] 760 [80315 109778] I [5 7] [64331 84531] 760 [84686 110911] I [7 10] [637036 900637] 760 [83860 118560] In Table 1 if I = 0 it is clear that the neutrosohic number otimization roblem is degenerated to the cris otimization roblem (ie traditional determinate otimization roblem Then under a neutrosohic number environment neutrosohic number otimal results are changed as the indeterminate ranges are changed Therefore one will take some interval range of the indeterminacy I in actual alications to satisfy actual indeterminate requirements of the truss structure design For examle if we take the indeterminate range of I [0 1] then the neutrosohic number otimal solution is D =x 1 = [58737 79087] mm and H = x = 760mm In actual design we need the de-neutrosohication in the neutrosohic otimal solution to determinate the suitable otimal design values of the design variables to satisfy some indeterminate requirement For examle if we take the maximum values of the otimal solution for I [0 1] we can obtain D = 73mm and H = 760mm for the two-bar truss structure design to satisfy this indeterminate requirement However traditional otimization design methods [10-13] cannot exress and handle the otimization design roblems with neutrosohic number information and are Wenzhong Jiang Jun Ye Otimal Design of Truss Structures Using a Neutrosohic Number Otimization Model under
Neutrosohic Sets and Systems Vol 14 016 97 secial cases of the neutrosohic number otimization design method in some cases The comarison of the roosed neutrosohic number otimization design method with traditional otimization design methods demonstrates the usability and suitability of this neutrosohic number otimization design method under a neutrosohic number environment 4 Conclusion Based on the concets of neutrosohic numbers this aer defined the oerations of neutrosohic numbers and neutrosohic number functions to establish the neutrosohic number objective function and constraints in neutrosohic number otimization design roblems Then we roosed a general neutrosohic number otimization model with constrained otimizations for truss structure design roblems Next a two-bar truss structure design examle was rovided to illustrate the alication and effectiveness of the roosed neutrosohic number otimization design method However the indeterminate (neutrosohic number otimization roblems may contain indeterminate (neutrosohic number otimal solutions (usually neutrosohic numbers but not always which can indicate ossible otimal ranges of the design variables and objective function when indeterminacy I is secified as a ossible interval ranges in actual alications In general indeterminate designs usually imly indeterminate otimal solutions from an indeterminate viewoint Then in the de-neutrosohication satisfying actual engineering design requirements we can determinate the suitable otimal design values of design variables in the obtained otimal interval solution corresonding to designers attitudes and/or some risk situations to be suitable for actual indeterminate requirements It is obvious that the neutrosohic number otimization design method in a neutrosohic number environment is more useful and more suitable than existing otimization design methods of truss structures since the traditional determinate/indeterminate otimization design methods cannot exress and handle the neutrosohic number otimization design roblems under an indeterminate environment Therefore the neutrosohic number otimization design method rovides a new effective way for the otimal design of truss structures under indeterminate/neutrosohic number environments Nonetheless due to existing indeterminacy I in the neutrosohic number otimization model it may be difficult to solve comlex neutrosohic number otimization models In the future therefore we shall further study solving algorithms/methods for neutrosohic number otimization design roblems and aly them to mechanical and civil engineering designs under indeterminate / neutrosohic number environments References [1] F Smarandache Neutrosohy: Neutrosohic robability set and logic American Research Press Rehoboth USA 1998 [] F Smarandache Introduction to neutrosohic measure neutrosohic integral and neutrosohic robability Sitech & Education Publisher Craiova Columbus 013 [3] F Smarandache Introduction to neutrosohic statistics Sitech & Education Publishing 014 [4] J Ye Bidirectional rojection method for multile attribute grou decision making with neutrosohic numbers Neural Comuting and Alications (015 DOI: 101007/s0051-015-13-5 [5] J Ye Multile-attribute grou decision-making method under a neutrosohic number environment Journal of Intelligent Systems 5(3 (016 377-386 [6] LW Kong YF Wu J Ye Misfire fault diagnosis method of gasoline engines using the cosine similarity measure of neutrosohic numbers Neutrosohic Sets and Systems 8 (015 43-46 [7] J Ye Fault diagnoses of steam turbine using the exonential similarity measure of neutrosohic numbers Journal of Intelligent & Fuzzy Systems 30 (016 197 1934 [8] F Smarandache Neutrosohic recalculus and neutrosohic calculus EuroaNova Brussels 015 [9] J Ye R Yong QF Liang M Huang SG Du Neutrosohic functions of the joint roughness coefficient (JRC and the shear strength: A case study from the yroclastic rock mass in Shaoxing City China Mathematical Problems in Engineering Volume 016 (016 9 ages htt://dxdoiorg/101155/016/485709 [10] M Sonmez Artificial bee colony algorithm for otimization of truss otimization Alied Soft Comuting 11 (011 406 418 [11] SO Degertekin An imroved harmony search algorithms for sizing otimization of truss structures Comuters and Structures 9 93 (01 9 41 [1] S Jalili Y Hosseinzadeh A cultural algorithm for otimal design of truss structures Latin American Journal of Solids and Structures 1(9 (015 171-1747 [13] S Wang X Qing A mixed interval arithmetic/affine arithmetic aroach for robust design otimization with interval uncertainty Journal of Mechanical Design 138(4 (016 041403-041403-10 Received: December 9 016 Acceted: December 016 Wenzhong Jiang Jun Ye Otimal Design of Truss Structures Using a Neutrosohic Number Otimization Model under