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.

Accuracy Synthesis of Redundant Parallel Kinematic Mechanism with Weighted Least Square Method

2012 ◽  
Vol 499 ◽  
pp. 3-8
Author(s):  
Xin You Li ◽  
Wu Yi Chen
Keyword(s):  
Machine Tool ◽  
Conventional Method ◽  
Least Square Method ◽  
Least Square ◽  
Tolerance Allocation ◽  
Manufacturing Processes ◽  
Manufacturing Cost ◽  
Parallel Kinematic Mechanism ◽  
Parallel Kinematic ◽  
Kinematic Mechanism

In order to reduce manufacturing cost, a methodology of accuracy synthesis for machine tool was recommended by combining both machining cost and Least Square method. Weighted coefficients representing the machining difficulty of manufacturing processes were introduced. 3PRS/UPS redundant parallel kinematic mechanism (3PRS/UPS PKM) was taken as an example, and its component tolerances were derived by the proposed method. Comparing with conventional method, the component tolerances were allocated reasonably. A further tolerance allocation for spherical and rotational joints was studied in detail. And hence, the producibility of component was improved and the manufacturing cost was reduced. The results showed that the proposed method was capable of producing tolerance allocations economically and accurately.

An Immune Algorithm for Least Cost Advanced Tolerance Design Problem

2006 ◽  
Vol 505-507 ◽  
pp. 511-516
Author(s):  
Ta Cheng Chen ◽  
Tung-Chou Hsu
Keyword(s):  
Immune Algorithm ◽  
Allocation Problem ◽  
Tolerance Allocation ◽  
Mixed Integer ◽  
Manufacturing Cost ◽  
Process Selection ◽  
Artificial Immune ◽  
Artificial Immune Algorithm ◽  
Allocation Problems ◽  
Artificial Immune Algorithms

This paper considers nonlinearly mixed integer tolerance allocation problems in which both tolerance and process selection are to be decided simultaneously so as to minimize the manufacturing cost. The tolerance allocation problem has been studied in the literature for decades, usually using mathematical programming or heuristic/metaheuristic optimization approaches. The difficulties encountered for both methodologies are the number of constraints and the difficulty of satisfying the constraints. A penalty-guided artificial immune algorithm is presented for solving such mixed integer tolerance allocation problems. Numerical examples indicate that the proposed artificial immune algorithms perform well for the tolerance allocation problem considered in this paper. In particular, as reported, solutions obtained by artificial immune algorithm are as well as or better than the previously best-known solutions.

scholarly journals Improved Sampling of the Pareto-Front in Multiobjective Genetic Optimizations by Steady-State Evolution: A Pareto Converging Genetic Algorithm

2002 ◽  
Vol 10 (3) ◽  
pp. 283-314 ◽  
Author(s):  
Rajeev Kumar ◽  
Peter Rockett
Keyword(s):  
Genetic Algorithm ◽  
Steady State ◽  
Pareto Front ◽  
Genetic Material ◽  
Solution Space ◽  
Computational Effort ◽  
Benchmark Problems ◽  
Multiobjective Genetic Algorithms ◽  
State Evolution ◽  
Real World Problems

Previous work on multiobjective genetic algorithms has been focused on preventing genetic drift and the issue of convergence has been given little attention. In this paper, we present a simple steady-state strategy, Pareto Converging Genetic Algorithm (PCGA), which naturally samples the solution space and ensures population advancement towards the Pareto-front. PCGA eliminates the need for sharing/niching and thus minimizes heuristically chosen parameters and procedures. A systematic approach based on histograms of rank is introduced for assessing convergence to the Pareto-front, which, by definition, is unknown in most real search problems. We argue that there is always a certain inheritance of genetic material belonging to a population, and there is unlikely to be any significant gain beyond some point; a stopping criterion where terminating the computation is suggested. For further encouraging diversity and competition, a nonmigrating island model may optionally be used; this approach is particularly suited to many difficult (real-world) problems, which have a tendency to get stuck at (unknown) local minima. Results on three benchmark problems are presented and compared with those of earlier approaches. PCGA is found to produce diverse sampling of the Pareto-front without niching and with significantly less computational effort

Optimal Tolerance Allocation Over Multiple Manufacturing Alternatives

1992 ◽  
Author(s):  
Jonathan Cagan ◽  
Thomas R. Kurfess
Keyword(s):  
Simulated Annealing ◽  
Optimization Problem ◽  
Simulated Annealing Algorithm ◽  
Minimum Cost ◽  
Tolerance Allocation ◽  
Hyperbolic Function ◽  
Manufacturing Processes ◽  
Worst Case ◽  
Annealing Algorithm ◽  
Manufacturing Alternatives

Abstract We introduce a methodology for concurrent design that considers the allocation of tolerances and manufacturing processes for minimum cost. Cost is approximated as a hyperbolic function over tolerance, and worst-case stack-up tolerance is assumed. Two simulated annealing techniques are introduced to solve the optimization problem. The first assumes independent, unordered, manufacturing processes and uses a Monte-Carlo simulation; the second assumes well known individual process cost functions which can be manipulated to create a single continuous function of cost versus tolerance with discontinuous derivatives solved with a continuous simulated annealing algorithm. An example utilizing a system of friction wheels over the manufacturing processes of turning, grinding, and saw cutting bar stock demonstrates excellent results.

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.

Optimum tolerance synthesis of simple assemblies with nominal dimension selection using genetic algorithm

2016 ◽  
Vol 230 (19) ◽  
pp. 3488-3508 ◽  
Author(s):  
D Vignesh Kumar ◽  
D Ravindran ◽  
M Siva Kumar ◽  
MN Islam
Keyword(s):  
Genetic Algorithm ◽  
Vital Role ◽  
Tolerance Allocation ◽  
Manufacturing Cost ◽  
Test Results ◽  
Manufacturing Costs ◽  
Critical Dimensions ◽  
Optimal Adjustment ◽  
Selection Of ◽  
Nominal Dimension

Optimum tolerance allocation plays a vital role in minimization of the direct manufacturing cost, and it is sensitive to tolerances related to variations in manufacturing processes. However, optimal adjustment of both nominal dimensions and selection of tolerances may further reduce assembly manufacturing cost and wastage of materials during processing. Most studies in existing literature focus on optimum tolerance allocation for the assemblies without considering nominal dimension selection. The method proposed in this work uses genetic algorithm techniques to allocate tolerances to assembly components, thereby minimizing costs. The component alternate nominal dimensions are predicted based on critical dimensions and its tolerances. The effectiveness of the developed algorithms demonstrated using randomly generated problems as well as sample problems taken from the literature. Test results are compared with those obtained using the Lagrange multiplier method. It is shown that by adjusting the nominal dimensions, the proposed method yields considerable savings in manufacturing costs.

Taking the Grunt Work Out of Tolerance Optimization

2006 ◽  
Author(s):  
Christopher Jayakaran ◽  
Ragini Patel ◽  
Prashant Momaya ◽  
K. Roopesh ◽  
Umeshchandra Ananthanarayana ◽  
...  
Keyword(s):  
Optimization Problem ◽  
Optimal Allocation ◽  
Tolerance Allocation ◽  
Manufacturing Processes ◽  
Process Selection ◽  
Second Stage ◽  
Product Design Process ◽  
Stage Process ◽  
Manufacturing Process Selection ◽  
Selection Of

The activity of tolerance allocation and optimization is a critical step in the product design process. This inherent trade-off between design objectives and process capability poses challenges in achieving right tolerances, both technically and effort-wise. Traditional methods in tolerance allocation are mostly regressive and are constrained by selection of the manufacturing processes. A progressive approach to tolerance allocation that does not assume these processes helps in achieving optimality of the tolerances and selection of manufacturing processes to realize the design. The two-stage process suggested in this paper formulates an optimization problem that allocates the tolerances based on sensitivities of tolerance values at the first stage followed by manufacturing process selection and further optimization to adhere to the processes selected in the second stage. The approach aims at achieving optimal allocation of tolerances and assignment of the manufacturing processes, while keeping the optimization problem computationally simple, although iterative.

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.

scholarly journals Sampling Scenario Set Partition Dual Bounds for Multistage Stochastic Programs

2020 ◽  
Vol 32 (1) ◽  
pp. 145-163 ◽  
Author(s):  
Ilke Bakir ◽  
Natashia Boland ◽  
Brian Dandurand ◽  
Alan Erera
Keyword(s):  
Optimization Problem ◽  
Computational Effort ◽  
Scenario Tree ◽  
Benchmark Problems ◽  
Set Partition ◽  
Random Parameters ◽  
Stochastic Programs ◽  
Require Solution ◽  
Multistage Stochastic Programs ◽  
Dual Bounds

We consider multistage stochastic programming problems in which the random parameters have finite support, leading to optimization over a finite scenario set. There has been recent interest in dual bounds for such problems, of two types. One, known as expected group subproblem objective (EGSO) bounds, require solution of a group subproblem, which optimizes over a subset of the scenarios, for all subsets of the scenario set that have a given cardinality. Increasing the subset cardinality in the group subproblem improves bound quality, (EGSO bounds form a hierarchy), but the number of group subproblems required to compute the bound increases very rapidly. Another is based on partitions of the scenario set into subsets. Combining the values of the group subproblems for all subsets in a partition yields a partition bound. In this paper, we consider partitions into subsets of (nearly) equal cardinality. We show that the expected value of the partition bound over all such partitions also forms a hierarchy. To make use of these bounds in practice, we propose random sampling of partitions and suggest two enhancements to the approach: sampling partitions that align with the multistage scenario tree structure and use of an auxiliary optimization problem to discover new best bounds based on the values of group subproblems already computed. We establish the effectiveness of these ideas with computational experiments on benchmark problems. Finally, we give a heuristic to save computational effort by ceasing computation of a partition partway through if it appears unpromising.

Top-Down Decomposition of Multi-Product Requirements Onto Locator Tolerances

2007 ◽  
Author(s):  
Johan Lo¨o¨f ◽  
Rikard So¨derberg
Keyword(s):  
The Body ◽  
Allocation Problem ◽  
Tolerance Allocation ◽  
Manufacturing Cost ◽  
Top Down ◽  
Complex Assembly ◽  
Optimal State ◽  
Selection Of ◽  
Rear Lamp

The tolerance allocation problem consists of choosing tolerances on dimensions of a complex assembly so that they combine into an ‘optimal state’ while fulfilling certain requirements on an allowed variation. This optimal state often coincides with the minimum manufacturing cost of the product. Sometimes it is balanced with an artificial cost that the deviation from target induces on the quality of the product. This paper suggests a multiobjective formulation of the tolerance allocation problem to automatically decompose requirements for an allowed variation on a set of critical product dimensions. This formulation is demonstrated using a rear lamp on a car with multiple requirements on allowed variation. In this case only the tolerances on locators that locates the lamp on the body are considered. The paper also reviews a selection of work that has been made on solving tolerance allocation problems.

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