ارزیابی تحمل صاف حداقل ناحیه با استفاده از تکنیک های برنامه ریزی غیرخطی: یک رویکرد گسترده

In general, non-linear optimization programs are formulated to evaluate the minimum zone straightness. This paper presents a spreadsheet approach that can be applied to determine the straightness errors of discrete data sampled from a continuous shape. The developed approach is easy to implement, and can obtain the minimum zone straightness based on the international standard, ANSI Y14.5M standard on geometric dimensioning and tolerancing. The primary goal of this spreadsheet implementation attempts to help reduce the possibility of making erroneous inspection decision, and then to precisely reflect the effect of inspection as early as possible for the purpose of quality control. An experimental study is conducted on examples taken from the literature and simulation data sets. Comparisons of the proposed approach against the existing methods in the previous studies are reported. Furthermore, the approach is demonstrated to be a viable tool for straightness verification in terms of the simulation data sets in which the theoretical straightness errors are known as a priori.

Measurement is commonly used to examine the quality of manufactured components against the established standards and specifications. Generally, the features of a manufactured part deviate in size and form. The accuracy of the size and form has a significant effect on the function of the final assembly. The modern manufacturing is characterized by the use of interchangeable parts produced with necessarily slight variation to ensure that they are of functional equivalence. The measurement of straightness for manufactured parts is one of the most frequently used procedures in metrology. With the increasing demand of the manufacturing automation, the high-speed measurement instruments such as the coordinate measuring machines (CMMs), machine vision, optical interferometer, etc., have been developed for this purpose. The computer is generally used for movement control of the instruments and tolerance evaluation of measured data. With the wide acceptance and applications of coordinate measuring instruments in practice, at present manufacturing engineers face a new challenge for the evaluation of geometric tolerances. The reason is that most instruments used in industry depend on discrete points to measure the specified dimensions and tolerances. The measured data do not give a direct assessment of form tolerance. Coordinate measuring systems have emerged to be important straightness verification tools owing to the recent advancements in computerized numerical control and precision machining. However, coordinate measuring systems still encounter difficult problems (Walker, 1988) such as correctly and unambiguously interpreting the definition of tolerances given in ANSI Y14.5M (1982) standard, formulating the problem of form error evaluation precisely as optimization models (particularly non-linear programs), and developing assessment algorithms which are consistent with ANSI Y14.5M standard, highly efficient, robust, and easy to use. It is necessary to apply a tolerance evaluation algorithm to interpret the continuous part features from the discrete measured coordinates. To evaluate the straightness errors from the measured points of the work piece surface, the ideal lines (substitute features) have to be established from the actual measurement satisfying the requirements defined in the standard, ANSI Y14.5M. The straightness error is then defined as the maximum peak-to-valley distance from the ideal features. The ANSI Y14.5M standard provides requirements for dimensioning and tolerancing. However, it gives little direction concerning the establishment of the ideal features. The least-squares method (LSM), which minimizes the sum of squared errors, is most widely used in the metrology community due to its computational simplicity and solution uniqueness. The LSM is only capable of obtaining an approximate solution that does not guarantee the requirements mentioned earlier. Furthermore, the LSM can result in a possible overestimation of the straightness error and the rejection of good products. During the past decade, some researchers have developed various methods to verify the minimum zone straightness. The issue of minimum zone verification has been discussed comprehensively by Murthy and Abdin (1980). In addition, Murthy and Abdin proposed and compared several methods such as the Monte Carlo method (MCM), simplex method (SPM) and spiral search method (SSM) to evaluate straightness errors. Shunmugan, 1991, Shunmugan, 1986, Shunmugan, 1987a and Shunmugan, 1987b presented various search procedures such as median technique (MDT), minimum deviation (MID), minimum average deviation (MAD), SPM and minimum zone line (MZL) to evaluate the straightness errors. Traband, Joshi, Wysk, and Cavalier (1989) developed a methodology based on the concept of convex hull zone (CHZ) to evaluate straightness errors of measured coordinates from a CMM. Huang, Fan, and Wu (1993) introduced a minimum zone method, namely control line rotation scheme (CLRS) for the straightness analysis. This method is applied to straightness analysis by rotations of the enclosing lines in half-field only. Kanada and Suzuki (1993) discussed the application of five computing techniques for evaluating straightness errors. They compared the solutions obtained by SPM, linear search method with quadratic interposition (QIM), linear search method with golden section (GSM), linearized method (TKM) and mixed method of the above-mentioned TKM and QIM methods. Carr and Ferreira (1995) proposed an algorithm, which solves a sequence of linear programs (SLP) to converge the solution of a non-linear program for the minimum zone straightness. Cheraghi, Lim, and Motavalli (1996) developed an optimization technique zone (OTZ) method based on the linear search to calculate the straightness error. Carr and Ferreira and Cheraghi et al. also presented the comparisons of several previous verification algorithms by using a set of examples. Current verification algorithms for coordinate measuring systems is based on least-squares solution, which minimizes the sum of squared errors, resulting in a possible overestimation of the form tolerance. Instead of using non-linear programs with normal deviations in straightness, researchers and practitioners usually utilized linear deviations and linear programs (Traband et al., 1989). Assessment algorithms developed in previous studies apply linear approximation approaches which can give incorrect results to non-linear form-fitting problems (Phillips, Borchardt, & Gaskey, 1993). The minimum zone straightness can be formulated as a non-linear optimization problem for accuracy purpose. As the functional requirements of products become more complicated and the tolerances become more stringent, measurement is one of the fundamental concerns in quality control. Rather than LSM, a number of tolerance evaluation methods have been extensively studied in the literature. Yet, they are rarely applied in the industry primarily due to the complexity of the associated calculations. To increase the accuracy and efficiency of tolerance verification, this paper presents a simple spreadsheet application for evaluating minimum zone straightness. The method proposed in this paper needs proper reformulation and cell allocations adapted in the spreadsheet, and then a newer generalized reduced gradient method (GRG2) embedded into Microsoft EXCEL™ will be automatically triggered to minimize the straightness error of measured data from a coordinate measuring instrument. The developed evaluation approach is an easy-to-implement one, and can obtain the exact straightness errors based on the international standard, ANSI Y14.5M standard on geometric dimensioning and tolerancing. In the remainder of the paper, Section 2 introduces the mathematical formulation of minimum zone straightness. Section 3 presents the proposed spreadsheet approach for straightness assessment. In Section 4, existing data and simulation data are adopted to verify the effectiveness of the proposed assessment method. Section 5 concludes this paper.

An alternative approach to the minimum zone straightness problem based on the GRG algorithm was presented. The straightness problem was formulated as an NLP problem. A GRG-based solution procedure was implemented in the widely accepted spreadsheet environment, EXCEL, to solve this NLP problem. It was shown through seven numerical examples and several simulation tests that the spreadsheet approach proposed in this article permits a straightforward formulation of the straightness evaluation problem, and always renders accurate information of straightness errors to mirror the manufacturability of processes under investigation. In comparison to the existing methods, the proposed procedure also has a competing edge on anchoring robust and stable solutions for larger straightness problem. From the results presented in Section 4, the GRG-based spreadsheet procedure for straightness evaluation agrees quite well to the ANSI Y14.5M standard on dimensioning and tolerancing. Moreover, the availability and portability of GRG codes make this method relatively easy to implement in practice for industrial users. Engineers applying this spreadsheet approach do not require additional optimization knowledge but effortless spreadsheet manipulation.