Optimal Statistical Tolerance Allocation of Assemblies for Minimum Manufacturing Cost

2011 ◽  
Vol 52-54 ◽  
pp. 1818-1823 ◽  
Author(s):  
Kuo Ming Cheng ◽  
Jhy Cherng Tsai
Keyword(s):  
Optimization Problem ◽  
Tolerance Allocation ◽  
Lambert W Function ◽  
Closed Form Expression ◽  
Manufacturing Cost ◽  
Form Expression ◽  
Algorithmic Approach ◽  
Constrained Minimization Problems ◽  
Tolerance Model ◽  
Statistical Tolerance

This paper explores a systematic method for optimal statistical tolerance allocation using the Lagrange multiplier method for minimizing manufacturing cost subject to constraints on dimensional chains and machining capabilities. The reciprocal power and exponential cost-tolerance models for statistical tolerancing are investigated for employing this method. The optimization problem is solved by applying the algorithmic approach. Especially, we further derive a closed-form expression of the tolerance optimization problem for reciprocal exponential cost-tolerance model by introducing the Lambert W function. For constrained minimization problems with only equality constraints, the optimum tolerance allocation can be obtained by solving simultaneous equations without further differentiating. An example is illustrated to demonstrate this approach. The result also shows that tolerances can be allocated quickly, economically and accurately using this method.

A Closed-Form Approach for Optimum Tolerance Allocation of Assemblies with General Tolerance-Cost Function

2011 ◽  
Vol 201-203 ◽  
pp. 1272-1278
Author(s):  
Kuo Ming Cheng ◽  
Jhy Cherng Tsai
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Equality Constraints ◽  
Allocation Problem ◽  
Tolerance Allocation ◽  
Lambert W Function ◽  
Manufacturing Cost ◽  
Worst Case ◽  
Constrained Minimization Problems

Tolerancing is one of the most crucial foundations for industry development and an index of product quality and cost. As tolerance allocation is based on manufacturing costs, this paper proposes a comprehensive method for optimal tolerance allocation with minimum manufacturing cost subject to constraints on dimensional chains and machining capabilities. The general reciprocal power and exponential cost-tolerance models with equality constraints as well as the worst-case and statistical tolerancings are employed in this method. A closed-form solution for the optimization problem by applying Lagrange multipliers is derived. The optimal tolerance allocation problem for reciprocal exponential cost-tolerance model by introducing Lambert W function is demonstrated. For constrained minimization problems with only equality constraints, the optimum design can be obtained by solving simultaneous equations without differentiating. An example is illustrated to demonstrate this approach. The result also shows that tolerance can be allocated economically and accurately using this method. The contribution of this paper is to solve the optimal tolerancing allocation problem by an efficient and robust method with simultaneous active constraints.

The Reduced Tolerance Allocation Problem

2016 ◽  
Author(s):  
David Sh. L. Shoukr ◽  
Mohamed H. Gadallah ◽  
Sayed M. Metwalli
Keyword(s):  
Genetic Algorithm ◽  
Optimization Problem ◽  
Computational Effort ◽  
Allocation Problem ◽  
Tolerance Allocation ◽  
Benchmark Problems ◽  
Manufacturing Processes ◽  
Manufacturing Cost ◽  
Functional Requirements ◽  
Different Dimensions

Tolerance allocation is a necessary and important step in product design and development. It involves the assignment of tolerances to different dimensions such that the manufacturing cost is minimum, while maintaining the tolerance stack-up conditions satisfied. Considering the design functional requirements, manufacturing processes, and dimensional and/or geometrical tolerances, the tolerance allocation problem requires intensive computational effort and time. An approach is proposed to reduce the size of the tolerance allocation problem using design of experiments (DOE). Instead of solving the optimization problem for all dimensional tolerances, it is solved for the significant dimensions only and the insignificant dimensional tolerances are set at lower control levels. A Genetic Algorithm is developed and employed to optimize the synthesis problem. A set of benchmark problems are used to test the proposed approach, and results are compared with some standard problems in literature.

Closed-Form Optimal Tolerance for Minimum Manufacturing Cost and Quality Loss Cost

2013 ◽  
Vol 655-657 ◽  
pp. 2084-2087 ◽  
Author(s):  
Shao Gang Liu ◽  
Qiu Jin
Keyword(s):  
Closed Form ◽  
Lagrange Multiplier ◽  
Significant Influence ◽  
Tolerance Allocation ◽  
Manufacturing Cost ◽  
Functional Requirement ◽  
Quality Loss ◽  
Functional Constraints ◽  
Power Cost ◽  
Tolerance Model

Tolerance allocation have significant influence on the manufacturing cost and quality loss cost. In order to obtain optimal tolerance, Lagrange multiplier method is used to minimize the summation of manufacturing cost and quality loss cost subject to constraints on product functional requirement. The reciprocal power cost-tolerance model with different functional constraints is considered, and closed-form optimal tolerances are obtained. Using the model proposed in this paper, the optimal tolerance can be obtained quickly and accurately. One example is used to illustrate the method proposed in this paper.

scholarly journals Synthesizing of Planar and Conformal Double-Sided Parallel-Cross Dipoles FSS Based on Closed-Form Expression

IEEE Access ◽  
2021 ◽  
pp. 1-1
Author(s):  
Yassine Zouaoui ◽  
Larbi Talbi ◽  
Khelifa Hettak ◽  
Naresh K. Darimireddy
Keyword(s):  
Closed Form ◽  
Closed Form Expression ◽  
Form Expression

Distribution of Consensus in a Broadcasting-based Consensus-forming Algorithm

2021 ◽  
Vol 48 (3) ◽  
pp. 91-96
Author(s):  
Shigeo Shioda
Keyword(s):  
Distribution Function ◽  
Probability Density Function ◽  
Closed Form ◽  
Probability Density ◽  
Infinite Number ◽  
Stable Distribution ◽  
Random Variable ◽  
Cauchy Distribution ◽  
Closed Form Expression ◽  
Form Expression

The consensus achieved in the consensus-forming algorithm is not generally a constant but rather a random variable, even if the initial opinions are the same. In the present paper, we investigate the statistical properties of the consensus in a broadcasting-based consensus-forming algorithm. We focus on two extreme cases: consensus forming by two agents and consensus forming by an infinite number of agents. In the two-agent case, we derive several properties of the distribution function of the consensus. In the infinite-numberof- agents case, we show that if the initial opinions follow a stable distribution, then the consensus also follows a stable distribution. In addition, we derive a closed-form expression of the probability density function of the consensus when the initial opinions follow a Gaussian distribution, a Cauchy distribution, or a L´evy distribution.

scholarly journals Colored HOMFLY-PT for hybrid weaving knot $$ {\hat{\mathrm{W}}}_3 $$(m, n)

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Vivek Kumar Singh ◽  
Rama Mishra ◽  
P. Ramadevi
Keyword(s):  
Closed Form ◽  
Topological Strings ◽  
Chebyshev Polynomials ◽  
Fibonacci Numbers ◽  
Infinite Family ◽  
Closed Form Expression ◽  
Two Dimensional ◽  
Laurent Polynomials ◽  
Form Expression ◽  
Torus Knots

Abstract Weaving knots W(p, n) of type (p, n) denote an infinite family of hyperbolic knots which have not been addressed by the knot theorists as yet. Unlike the well known (p, n) torus knots, we do not have a closed-form expression for HOMFLY-PT and the colored HOMFLY-PT for W(p, n). In this paper, we confine to a hybrid generalization of W(3, n) which we denote as $$ {\hat{W}}_3 $$ W ̂ 3 (m, n) and obtain closed form expression for HOMFLY-PT using the Reshitikhin and Turaev method involving $$ \mathrm{\mathcal{R}} $$ ℛ -matrices. Further, we also compute [r]-colored HOMFLY-PT for W(3, n). Surprisingly, we observe that trace of the product of two dimensional $$ \hat{\mathrm{\mathcal{R}}} $$ ℛ ̂ -matrices can be written in terms of infinite family of Laurent polynomials $$ {\mathcal{V}}_{n,t}\left[q\right] $$ V n , t q whose absolute coefficients has interesting relation to the Fibonacci numbers $$ {\mathrm{\mathcal{F}}}_n $$ ℱ n . We also computed reformulated invariants and the BPS integers in the context of topological strings. From our analysis, we propose that certain refined BPS integers for weaving knot W(3, n) can be explicitly derived from the coefficients of Chebyshev polynomials of first kind.

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