Tolerance evaluation of minimum zone straightness using non-linear programming techniques: a spreadsheet approach

Computers & Industrial Engineering 43 (2002) 437±453

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Tolerance evaluation of minimum zone straightness using non-linear programming techniques: a spreadsheet approach Mu-Chen Chen a, Shu-Kai S. Fan b,* a

Institute of Commerce Automation and Management, National Taipei University of Technology, 1, Sec. 3, Chung-Hsiao E. Road, Taipei 106, Taiwan, ROC b Department of Industrial Engineering and Management, Yuan Ze University, 135 Yuan-Tung Road, Chung-Li, Taoyuan 320, Taiwan, ROC

Abstract 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 re¯ect 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 veri®cation in terms of the simulation data sets in which the theoretical straightness errors are known as a priori. q 2002 Elsevier Science Ltd. All rights reserved. Keywords: Tolerance evaluation; Minimum zone straightness; Generalized reduced gradient (GRG) methods; Constrained optimization

1. Introduction Measurement is commonly used to examine the quality of manufactured components against the established standards and speci®cations. Generally, the features of a manufactured part deviate in size and form. The accuracy of the size and form has a signi®cant effect on the function of the ®nal 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 * Corresponding author. Tel.: 1886-3-463-8800x510; fax: 1886-3-463-8907. E-mail address: [email protected] (S.-K.S. Fan). 0360-8352/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S0 3 6 0 - 8 3 5 2 ( 0 2 ) 0 0 05 7 - 8

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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 speci®ed dimensions and tolerances. The measured data do not give a direct assessment of form tolerance. Coordinate measuring systems have emerged to be important straightness veri®cation tools owing to the recent advancements in computerized numerical control and precision machining. However, coordinate measuring systems still encounter dif®cult problems (Walker, 1988) such as correctly and unambiguously interpreting the de®nition 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 ef®cient, 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 de®ned in the standard, ANSI Y14.5M. The straightness error is then de®ned 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 veri®cation 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 (1986, 1987a,b, 1991) 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-®eld only. Kanada and Suzuki (1993) discussed the application of ®ve 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

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439

Fig. 1. An illustrative example of minimum zone straightness.

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 veri®cation algorithms by using a set of examples. Current veri®cation 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-®tting 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 ef®ciency of tolerance veri®cation, 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 EXCELe 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. 2. Mathematical formulation As de®ned by the ANSI Y14.5M standard, straightness is a condition for which an element of a surface is a straight line. Minimum zone straightness speci®es a tolerance zone de®ned by two parallel lines that contain all measured coordinates from the work piece surface. Fig. 1 shows the geometric relationship of this minimum zone condition. In this case, the sampled pro®le should be contained between two parallel straight lines 0.02 apart. The measurement of straightness is a 2D measurement. It should be noted that the straight feature of a surface has no size.

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Fig. 2. Minimum zone straightness after translation and rotation.

Mathematically speaking, for an element of a surface, minimum zone straightness can be de®ned as follows. Given a set of measured coordinates in two dimensions, P ˆ {pi ˆ …xi ; yi †; i ˆ 1; 2; ¼; N} for straightness evaluation, it is possible to ®nd explicitly two ~ parallel lines minimum distance apart that enclose all points. Since the two parallel lines could be located anywhere in a plane, the data set P is translated along an arbitrary axis and/or rotated around an axis. The relative positions of the ~measured points in the set with each other will not shift. For simpli®cation, it is assumed that one of the two lines de®ning the minimum zone coincides with the x-axis (i.e. y ˆ 0) (Fig. 2), the other is parallel to the ®rst line at a distance ts from it (i.e. y ˆ ts ). Hence, the de®nition of the minimum zone straightness can be rewritten as: given two parallel lines separated by a distance of ts ; in which all points are enclosed and the zone is minimized. Let P 0 ˆ {pi 0 ˆ …xi 0 ; yi 0 †; i ˆ 1; 2; ¼; N} be the new set of measured points ~ after they are rotated through an angle u , and translated by X and Y along x-axis and y-axis, respectively. Then the new positions can be expressed as … x 0i

T y 0i † ˆ

cos u

2sin u

sin u

cos u

! … xi

yi †T 1 … X

Y †T

…1†

According to the above-mentioned assumptions, the exact value of straightness error may be found by solving the following mathematical model Minimize ts

s:t: y 0i # ts ; y 0i $ 0; i ˆ 1; 2; ¼; N

…2†

In terms of Eq. (1), problem (2) can be expressed as the following non-linear optimization program: Minimize ts s:t: xi sin u 1 yi cos u 1 Y # ts ; xi sin u 1 yi cos u 1 Y $ 0; i

…3†

ˆ 1; 2; ¼; N; Y; u unrestrictedinsign …URS† It is important to note that x-coordinate (locations on the work piece surface where the measurements are conducted) represents the independent variable, since the straight feature of a surface has no size and the minimum zone is de®ned by the lines parallel to the x-axis.

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3. The solution procedure with the GRG algorithm 3.1. Motivation for using the GRG method The straightness problem as in Eq. (3) is a non-linear programming (NLP) problem since the 2N constraints belong to functions of trigonometry, even the objective function being linear. There is a wide variety of NLP methods/software for solving these types of problems. We chose the generalized reduced gradient (GRG) method in NLP for the following reasons. (i) In instances where the derivative information of the objective function and constraints is at hand, the gradient-based solution procedures will be more trustworthy than derivative-free methods recommended by various researchers aforementioned, since (at least) the ®rst-order necessary condition (FONC) for local optimality is satis®ed. Problem (3) can solely be deemed as a deterministic optimization program provided repeatability and reproducibility (R&R) is warranted based primarily on the wellcalibrated equipment and skilled personnel, respectively. Perhaps under particular circumstances where the inherent variability while taking measurements is negligible, the computed solution to problem (3) can be trusted with con®dence. Hence, the class of gradient-based search is favored for this situation. Note that most of direct search procedures are essentially devised for unconstrained optimization problems (Bazaraa, Sherali, & Shetty, 1993). (ii) The family of GRG-based algorithms is often referred to as a primal method where primal feasibility is maintained during the optimization process (Luenberger, 1989). This indicates that the method searches through the feasible region (bounded by the constraints) for the optimal point. Each point tested during optimization is feasible and for each iteration, the value of the objective function sequentially improves. Since the incumbent solution at each iteration is feasible, the solution point returned by the last iteration halting the algorithm is feasible. Therefore, even though for general problems the GRG method is not guaranteed to provide `global' optima without performing the ad hoc analysis, the termination point is feasible and has a much better objective function value than the starting point. This is in perfect agreement with our initial intention (from the aspect of precision engineering), that is, the search for a feasible solution with an acceptable value of minimum zone straightness is desired rather than the achievement of global optimality. (iii) The GRG algorithm has been proved one of the most robust and ef®cient NLP methods available to solve small to moderate size problems (Lasdon, Waren, Jain, & Ratner, 1978; Reklaitis, Ravindran, & Ragsdell, 1983, chapter 12). (iv) Many GRG codes are available. The pioneering work can be traced back to Abadie (1972), and other researchers have developed their own GRG codes (Cohen, 1974; Heltne & Littschwager, 1973). GINO (Lasdon & Waren, 1990) is the GRG-based computer software that has been disseminated to a wide extent. It runs on any PC and is free of the student version upon the purchase of the popular OR text by Winston (1994). Moreover, the optimizer termed `SOLVER' embedded into the Microsoft EXCEL (1994) is the GRG2 code from Lasdon, Fox, and Ratner (1974) and Lasdon et al. (1978), and it will be used to illustrate our approach in Section 3.2. If measurement needs to be taken much extensively, problem (3) becomes dramatically large. However, the availability of GRG codes makes problem (3) easy to solve. On the other hand, the GRG2's portability via the use of spreadsheet solvers provides potential nontechnical users with a user-friendly interface and an easy implementation in practice. Note that global optima will de®nitely be attained if the initial point used to start the GRG algorithm is

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Fig. 3. The SOLVER's structure for the straightness problem.

suf®ciently close to the global solution known beforehand. Accordingly, the multi-search using different initial guesses will be an effective heuristic for GRG to locate the `best-practice' optimal solution. 3.2. Implementation of GRG in EXCEL The problem of minimum zone straightness is to seek the optimal parameters settings of transformation and translation while minimizing the straightness error. A detailed explanation of the GRG method is left out here since gaining a sound understanding of a NLP method is mathematically involved. For readers who are interested, a complete description can readily be accessed in public (Bazaraa et al., 1993). The solver add-in Dynamic Link Library (DLL, i.e. solver.dll or solver32.dll) gives EXCEL the power of linear and non-linear optimization. Solver not only ®nds a feasible solution to a problem, but also locates the (local) optimal solution, given a set of cells (representing decision variables) that it can change, a set of constraints that must be satis®ed, and one cell (denoting the objective function) that must be optimized for the minimum, maximum, or on-the-target. Next, a simple example with ®ve data points taken from Traband et al. (1989) is used to illustrate our spreadsheet approach by taking following steps. Step 1. First, use Tools Add-Ins to add the Solver and the Tools Solver command to run the Solver in Microsoft Windows. Step 2. Set up the cell-allocation and formulation for the problem as shown in Fig. 3. Here, the columns of x and y represent the measured coordinates of the data collected. The column of straightness indicates values of the straightness errors for each point during the optimization process. The column of optimal solution denotes the set of decision variables (Theta, Translation and Precision) that

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443

Fig. 4. The SOLVER dialog box for parameter setting.

SOLVER can change toward optimizing the objective function (Precision). Note that in problems of this kind, the objective function also serves as a decision variable. Step 3. Choose Tools Solver to start the program, and then SOLVER loads and the empty SOLVER parameters dialog box appear. In the Set Target Cell text box, reference the cell you want to optimize. For this example, select B11. De®ne the type of relation between the Set Target Cell and a solution value by selecting one of the following Equal To option buttons: Max, Min1 or Value. Thus, Min is selected for this problem. Select the By Changing Cells text box; then select the adjustable cells that SOLVER should change while attempting to ®nd the best answer. For this example, the cells are $B$9:$B$11 for which an initial starting guess can be given by selecting Guess if necessary. For this example, the input values to begin with are all 0s. Note that, in SOLVER, the defaulting setting of the initial guess for any decision variable is zero. To begin the GRG algorithm, a feasible starting point is needed. The point x ˆ 0 is certainly not feasible for problem (3) except some unexpected ideal cases. However, most GRG codes provide an initialization routine that will attempt to ®nd an initial starting point. Finally, choose Add to incorporate the 2N constraints as in Eq. (3) through designation of the corresponding cells de®ned in Step 2 into the list of constraints. After the SOLVER Parameters dialog box is completely ®lled in for

Fig. 5. Solution reports returned by SOLVER.

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Fig. 6. The ®nal worksheet for the straightness problem.

this problem, the cell to be optimized, the cells to be changed, and the constraints on the solution appear as shown in Fig. 4. Step 4. Tune up the GRG parameters in Options to be used in the optimization process if required. With respect to the details of the parameter setting and their effects, see the on-line Help on Windows. The default parameter setting in SOLVER is attached in Appendix A. Step 5. Having ®nished the parameter setting, click Solve to run SOLVER to ®nd the optimal solution. When SOLVER ®nds a solution, the Solver Results dialog box appears as shown in Fig. 5. Step 6. Select Keep Solver Solution and choose OK to keep the offered solution in the worksheet. The ®nal worksheet printout is shown in Fig. 6. In this dialog box, you can also choose the reports you may want to generate for the results of further analysis. 4. Computational experience 4.1. Numerical examples In this sub-section, various examples (Data Sets 1±7) found in the literature are revisited to verify the effectiveness of the proposed approach by means of the GRG algorithm for straightness evaluation. The ®rst ®ve data sets are from Traband et al. (1989), later utilized for a testing purpose in Cheraghi et al. (1996). The Data Sets 6 and 7 were also examined by Cheraghi et al. All the data points are listed in Appendix B.

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Table 1 The previously reported methods for Data Sets 1±7 Data Set

Methods

1

MDT, MAD, SPM, MZL (Shunmugan, 1986, 1987a, 1991), CHZ (Traband et al., 1989), OTZ (Cheraghi et al., 1996) CHZ (Traband et al, 1989), CLRS (Huang et al., 1993), OTZ (Cheraghi et al., 1996) CHZ (Traband et al, 1989), OTZ (Cheraghi et al., 1996) CHZ (Traband et al, 1989), OTZ (Cheraghi et al., 1996) CHZ (Traband et al, 1989), SLP (Carr & Ferreira, 1995), OTZ (Cheraghi et al., 1996) CLRS (Huang et al., 1993), OTZ (Cheraghi et al., 1996) MID (Shunmugan, 1987a), SLP (Carr & Ferreira, 1995)

2 3 4 5 6 7

The previously reported methods for analyzing Data Sets 1±7 are summarized in Table 1. For the Data Sets 1±7, the results of LSM and the best results presented in the previous studies are shown in Table 2 together with those generated by using the GRG method. All the solutions returned by the spreadsheet approach are, at least, as equally good as the best solutions ever found (Table 2). As mentioned previously, the LSM, is most widely used in the metrology community due to its computational simplicity and solution uniqueness. The LSM can result in a possible overestimation of the straightness error and the rejection of good products. As can be seen from Table 2, the spreadsheet approach returns optimal values that are signi®cantly smaller than those obtained by using LSM. Table 2 Straightness evaluation results for the Data Sets 1±7 Data Set

LSM

Best reported result

Spreadsheet approach (GRG)

1 2a 3 4 5 6 7

2.401 0.8877 0.1706 0.005377 0.001463 5.882 0.0028

2.1213 (CHZ, OTZ) 0.8578 (OTZ) 0.1646 (OTZ) 0.005186 (CHZ, OTZ) 0.001311 (CHZ, OTZ) 5.493 (OTZ) 0.002666 (SLP)

2.1213 0.8578 0.1645 0.005185 0.001311 5.493 0.002666

a

Cheraghi et al. (1996) reported that the result of Data Set 2 (0.8479) made by Traband et al. (1989) is erroneous.

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Table 3 Straightness tolerance results for L1

Noise (%) Maximum error Estimated error (ts)

Data Set 8

Data Set 9

Data Set 10

20 0.1 0.099816

40 0.1 0.099949

60 0.1 0.099715

4.2. Simulation study Nine sets of simulation data are produced from analytical lines and then analyzed so that the estimated straightness errors can be compared with the theoretical values. For each simulation data set, 50 pro®le points are generated. The points of these nine data sets (Data Sets 8±16) are created from the following lines: L1 : y 2 100 ˆ 0;

L2 : 24y 1 8 ˆ 0;

L3 : 2x 1 y 1 6 ˆ 0

Simulation data have various degrees of noise interference (Tables 3±5). The normal deviation between the simulated point and analytical line is generated randomly. In theory, the estimated straightness error equals the maximum deviation. The percentage of noise is the ratio of noisy data points to the total number of boundary points. From Tables 3±5, all estimated errors of simulation data sets are almost equal to the maximum deviations (theoretical errors), even though the noisy data points are as high as 60%. Based on the simulation study, the GRG method is effective for straightness evaluation. 4.3. Discussions As can be seen from Section 4.1, the computational results have demonstrated the effectiveness of the proposed procedure to provide `precise' values of straightness errors. In terms of several examples appearing in the literature, we found that the spreadsheet approach using GRG is superior to some of the existing methods and any other procedures produced, at best, equally good solutions. Based on the simulation results in Section 4.2, the GRG algorithm has been shown to be a stable solution procedure to solve the larger straightness evaluation problems. Nonetheless, the determination of computational times for SOLVER is not available since much of the optimization requires manual manipulation. A useful trick to accelerate the searching time is to provide SOLVER with a feasible starting point in that the default initial guess (where all control variables are set to zeroes) is not feasible to problem (3). Therefore, a trivial point with a value greater Table 4 Straightness tolerance results for L2

Noise (%) Maximum error Estimated error (ts)

Data Set 11

Data Set 12

Data Set 13

20 0.01 0.009969

40 0.01 0.009997

60 0.01 0.00991

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Table 5 Straightness tolerance results for L3

Noise (%) Maximum error Estimated error (ts)

Data Set 14

Data Set 15

Data Set 16

20 0.005 0.004947

40 0.005 0.00499

60 0.005 0.004944

or equal to the maximum of y among all the data points, assigned to ts will be a valid feasible initial guess. For example, when the feasible starting point, u ˆ Y ˆ 0 and ts ˆ 5; was used for the data set 1, SOLVER located the same solution more swiftly. 4.4. Implementation issues Add-in's extend EXCEL's capability by adding features, menus, and toolbars that work as though they were built in to EXCEL. If measuring minimum zone straightness becomes routine in common precision machining practice, for the utilization of the proposed method via GRG, it is wise to employ the SOLVER add-in ®le (i.e., solver.xla) shipped with EXCEL that helps the process engineer to use a variety of numeric methods for equation solving and optimization and that can be automatically called from Visual Basic for Microsoft Of®ce Applications (VBA). Appendix C gives a detailed look at these GRG sub-procedures that we can access from the SOLVER add-in. Hence, if an ad hoc VBA project is properly created, the volumes of straightness veri®cation work can be completed very swiftly and repetitively performed in batches. A crude implementation, where the computer programming language Visual Basic (VB) is used to build up a user interface for measurement data input, passes the data to a worksheet prepared in EXCEL (installed with solver.xla), automatically trigger SOLVER add-in via VBA programming for computing the minimum zone straightness and then pass the ®nal results back to the VB interface, is shown in Figs. 7 and 8 in terms of Data Set 2 in Appendix B. The measurement data can also be automatically fed into the computation module through machine±computer interface. A potential infrastructure of the successive (or batch-to-batch) minimum zone straightness evaluation system is shown in Fig. 9. 5. Concluding remarks 5.1. Summary 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

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Fig. 7. The data input interface for measuring minimum zone straightness.

448

449

Fig. 8. Final computational results of minimum zone straightness.

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Fig. 9. A projected scenario for automatically evaluating minimum zone straightness.

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. 5.2. Directions for future research An interesting opportunity for future research would be to build the interface using a website-based programming language which bridges the GRG algorithm in SOLVER and a general data input. If the shop ¯oor data is able to be real-time collected, then through the interface, the data can be universally captured by SOLVER via Internet and the straightness error evaluation can be on-site performed to achieve on-line quality control. On the other facet of future research, computational time (required by the algorithm to solve the straightness problem) becomes a major concern in the presence of the real-time inspection formerly addressed. To simplify matters, once the resulting value on the objective function passes the screen test prescribed by the speci®cation (e.g. denoted by tz), then the work piece or assembly being measured is considered as `conforming' or `To Go'. Apparently, this can be done by including one additional constraint, ts # tz ; into problem (3). For this situation, the use of the GRG method can be easily implemented in SOLVER by adding associated constraints or, conservatively, by selecting the objective function to Value option (i.e. ts ˆ tz ) but may be time-consuming from the aspect of algorithmic computing. Accordingly, the development of a meta-based heuristic that will be immediately terminated on condition that a feasible solution satisfying the speci®cation is found deserves further scrutiny. Appendix A See Table A1.

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451

Table A1 Default parameter setting in SOLVER Max time Max iterations Precision Tolerance Estimates Derivatives Search

100 s 100 0.000001 5% Tangent Forward Newtons

Appendix B See Table B1.

Table B1 Straightness data points Data Set 1

Data Set 2

Data Set 3

Data Set 4

Data Set 5

Data Set 6

Data Set 7

x

y

x

y

x

y

x

y

x

y

x

y

x

y

22 21 0 1 2

3 5 2 1 2

1 2 3 4 5 6 7 8 9 10

2.428 2.891 3.445 2.931 3.895 4.196 4.497 4.662 4.545 4.303

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00

20.0664500 20.0643800 0.0087610 20.0111700 20.0623700 20.0382900 0.0655000 0.0635700 0.0284900 20.0061130 20.0952500 20.0115400 20.0240600 0.0351500 20.0199700 0.0154000 20.0132400 20.0222500 0.0771000 20.0003596

0.3952 0.6953 0.9669 1.2762 1.5797 1.8593 2.1333 2.4197 2.6001 2.5890 3.0662 3.2165 3.4217 3.6179 3.8185

20.0032 20.0016 20.0042 20.0028 20.0037 20.0007 20.0010 0.0007 0.0007 0.0017 0.0025 20.0017 0.0026 0.0027 0.0047

0.2845 0.6600 1.2041 1.4994 1.8494 2.2261 2.5724 2.9076 3.2548 3.4142 3.6307 3.9237 4.2647 4.5122 4.8150 5.1334 5.3606 5.6534 5.9058 6.0774 6.2962 6.5240 6.7114 6.9996 7.2076

20.0034 20.0032 20.0030 20.0035 20.0036 20.0025 20.0028 20.0026 20.0031 20.0031 20.0029 20.0029 20.0028 20.0028 20.0027 20.0027 20.0030 20.0032 20.0020 20.0019 20.0019 20.0019 20.0017 20.0019 20.0017

0 10 20 30 40 50 60 70 80 90

0.0 2.0 3.0 2.0 2.5 21.0 2.0 5.0 6.0 3.0

250 225 0 25 50

0.003 0.005 0.002 0.001 0.002

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Table C1 SOLVER add-in sub-procedures that can be called from Visual Basic Sub-procedures Provided by SOLVER.XLA Add-in

Contents

SolverAdd(CellRef U , Relation U , FormulaText U ) SolverChange(CellRef U , Relation U , FormulaText U ) SolverDelete(CellRef U , Relation U , FormulaText U ) SolverFinish(KeepFinal U , ReportArray U ) SolverFinishDialog(KeepFinal U , ReportArray U ) SolverGet(TypeNum U , SheetName U ) SolverLoad(LoadArea U ) SolverOk(SetCell U , MaxMinVal U , ValueOf U , ByChange U ) SolverOkDialog(SetCell U , MaxMinVal U , ValueOf U , ByChange U ) SolverOptions(MaxTime U , Iterations U , Precision U , AssumeLinear U , StepThru U , Estimates U , Derivatives U , SearchOption U , IntTolerance U , Scaling U ) SolverReset SolverSave(SaveArea U ) SolverSolve(UserFinish U , ShowRef U )

Add the constraints Modify the existing constraints Delete the existing constraints De®ne the solving reports As above & show the dialog box Return the ®nal value speci®ed Load the existing SOLVER module De®ne the objective function and decision variables As above & show the dialog box GRG Parameter settings Reset to default Save the current SOVER module Execute the SOLVER program

Appendix C See Table C1. References Abadie, J. (1972). Application of the GRG algorithm to optimal control problems. In J. Abadie, Nonlinear and integer programming (pp. 191±211). Amsterdam: North-Holland. ANSI Y14.5M-1982 (1982). Dimensioning and tolerancing, New York: The American Society of Mechanical Engineers. Bazaraa, M. S., Sherali, H. D., & Shetty, C. M. (1993). Nonlinear programming: theory and algorithms, (2nd ed). New York: Wiley. Carr, K., & Ferreira, P. (1995). Veri®cation of form tolerances Part I: Basic issues, ¯atness and straightness. Precision Engineering, 17(2), 131±143. Cheraghi, S. H., Lim, H. S., & Motavalli, S. (1996). Straightness and ¯atness tolerance evaluation: An optimization approach. Precision Engineering, 18(1), 30±37. Cohen, C. (1974). Generalized reduced gradient technique for non-linear programming user writeup, Boston, MA: Vogelback Computing Center, Northeastern University. EXCEL 5.0a (1985±1993). Copyright q Microsoft Corporation. Heltne, D. R., & Littschwager, J. M. (1973). User's guide for GRG 73 and technical appendices to GRG 73, Iowa City, IA: College of Engineering, University of Iowa. Huang, S. T., Fan, K. C., & Wu, J. H. (1993). A new minimum zone method for evaluating straightness errors. Precision Engineering, 15(3), 158±165. Kanada, T., & Suzuki, S. (1993). Application of several computing techniques for minimum zone straightness. Precision Engineering, 15(4), 274±280. Lasdon, L. S., & Waren, A. D. (1990). GINO/PC, Copyright q 1984±89 Lasdon, L., & Waren, A. and LINDO Systems Inc., Portions Copyright q 1981 Microsoft Corporation. Lasdon, L. S., Fox, R. L., & Ratner, M. (1974). Nonlinear optimization using the generalized reduced gradient method. Reveu Francaise d' Automatique et Recherche Operationnelle, 23, 73±104.

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Lasdon, L. S., Waren, A. D., Jain, A., & Ratner, M. (1978). Design and testing of a generalized reduced gradient code for nonlinear programming. ACM Transactions on Mathematical Software, 4(1), 34±50. Luenberger, D. G. (1989). Linear and nonlinear programming, (2nd ed). Reading, MA: Addison-Wesley. Murthy, T. S. R., & Abdin, S. Z. (1980). Minimum zone evaluation of surfaces. International Journal of Tool Design and Research, 20(2), 123±136. Phillips, S. D., Borchardt, B., & Gaskey, G. (1993). Measurement uncertainty considerations for coordinate measuring machines, NISTIR 5170. Gaithersburg, MD: NIST. Reklaitis, G. V., Ravindran, A., & Ragsdell, K. M. (1983). Engineering optimization: Methods and applications, New York: Wiley. Shunmugan, M. S. (1986). On assessment of geometric errors. International Journal of Production Research, 24(2), 413±425. Shunmugan, M. S. (1987). Comparison of linear and normal deviations of forms of engineering surfaces. Precision Engineering, 9(2), 96±102. Shunmugan, M. S. (1987). New approach for evaluating form errors of engineering surfaces. Computer-Aided Design, 19(7), 368±374. Shunmugan, M. S. (1991). Establishing reference ®gures for form evaluation for engineering surfaces. Journal of Manufacturing Systems, 10, 314±321. Traband, M. T., Joshi, S., Wysk, S., & Cavalier, T. M. (1989). Evaluation of straightness and ¯atness tolerance using the minimum zone. Manufacturing Review, 2(3), 189±195. Walker, R. (1988). GIDEP Alert No. X1-A-88-01. Technical Report, Government-Industry Data Exchange Program, August 22. Winston, W. L. (1994). Operations research: Applications and algorithms, (3rd ed). Belmont, CA: Wadsworth.