Loss Function Cgg1

Author: Carlos González González Article: Taguchi Loss Function Condition: Nominal is Better Taguchi Loss Function detects the customer desire to produce products that are more homogeneous, piece a piece and low-cost product. The loss to society combines costs incurred, during production process, and costs found during use by the customer. Uniform products minimize the loss to society and reduce costs at the points of production and consumption. There is a comparison between the philosophy of goalpost syndrome Figure 1 and the loss function generated by a normal distribution centered with the target of specification see Figure 2.

good

bad

bad

bad

Figure 1. Goalpost Philosophy inside it is good outside it is bad Figure 2 shows that the quadratic loss function increase its value when the value of the items depart from target, showing too that the minimum cost stays on the target (Mean of Specification)

Figure 2. Quadratic Quality Loss Function and Probability Density Function (Normal Curve – Bell Shape). PhD. Genichi Taguchi uses the Mathematical equation (1-i): L = k(y – m)^2

(1-i)

In this equation, L is the loss associated with a particular value y, m is the nominal value of the specification, and the value of k is a constant depending on the cost at the specification limits and width of the specification. Applying this equation to the next problem we will have: Problem 2.27, page 55 of “Statistics” book of Murray R. Spiegel of McGraw-Hill Company Inc México, 2002. “The following table shows the diameters in centimeters of a sample of 60 ball bearings manufactured by a company. Construct a frequency distribution of the diameters, using appropriate class intervals.” Suppose that Limits of Specification are: LSL = 1.720 and USL = 1.760, and the cost of a rejected part at 1.720 and 1.760 is L = $1.00 Target or Mean of Specification = 1.740. What is the individual Loss for a ball of a value of 1.730 Table 1-i 1.738 1.728 1.745 1.733 1.735 1.732 1.729 1.737 1.736 1.730 1.732 1.737 1.743 1.736 1.742 1.732 1.735 1.731 1.740 1.735 1.740 1.730 1.727 1.746 1.736 1.724 1.728 1.739 1.734 1.735 1.741 1.733 1.738 1.734 1.732 1.735 1.735 1.742 1.725 1.738 1.736 1.729 1.731 1.736 1.733 1.739 1.741 1.734 1.726 1.739 1.734 1.727 1.736 1.730 1.737 1.735 1.732 1.735 1.744 1.740 Substituting values on equation (1-i) we have: $1.00 = k(LSL – m)^2 The lower specification limit (LSL) is substituted into the equation, which is where the $1.00 loss is incurred. The upper specification limit also could be used for this calculation. Solving for k, k = $1.00 / [(LSL – m)^2]

Giving that m = 1.740 in (nominal value), k = $1.00 / [(1.720 – 1.740)^2] k = $1.00 / [(– 0.02 )^2] k = $1.00 / (0.0004) k = 2,500 per cm^2 Therefore for a part with value of 1.730: L = 2,500(y – 1.740)^2

(1-ii)

L = 2,500(1.730 – 1.740)^2 L = 2,500(– 0.01)^2 L = 2,500(0.0001) = $0.25 A method of estimating average loss per part entails using the loss equation with a different form. Mathematically, this calculation is equivalent to using the average value of the (y – m)2 portion of the loss equation. Expanded, this is L = k[(y1 – m)^2 + (y2 – m)^2 + … + (yN – m)^2 +] / N

(1-iii)

Where N = number of parts sampled. If all the (y – m) values are squared, added together, and divided by the number of items, then the result is the desired value. For a large number of parts, the average loss per part is equal to. L = k[S^2+(ybar – m)^2] S^2 = variance around the average, ybar

(1-iv)

ybar = average value of y for the group. (ybar –m) = offset of the group average from the nominal value m Equations 1-iii and 1-iv are equivalent. For the set of values presented in Table 1-i the values of S^2 and ybar can be calculated by hand.

S^2 = (0.004951)^2 cm^2 = 0.000024512 ybar = 1.734867 Using Equation 1-iv we have: L = k[S^2+(ybar – m)^2] L = 2500[0.000024512 + (1.734867 – 1.740)^2] L = 2500[0.000024512 + (– 0.005133)^2] = 2500[0.000024512 + 0.000026347] L = 2500 x 0.000050859 = $ 0.1271475 per part We were working for an objective named “nominal is best”. Suppose that these 60 pieces were drawn from a lot of 5,000 parts the total loss will be the multiplication of the average loss per part times the total number of parts in the lot. Total Loss = 5,000 x 0.1271475 = $ 635.73 Because they are not produced with a value of target = 1.740. You can use the software in six different languages (Spanish, English, French, Deutsch, Italian and Portuguese) LOSSFUNCTION_EN from site: www.spc-inspector.com/cgg When we run the program first we need to build the file of data (60 measurements) when you click on [BUILDING FILE], giving the name including the termination .txt, then you need to open such file making click on [LOSS FUNCTION DATA OF FILE] and [NOMINAL IS BETTER] then you should provide the Limits of Specification and Loss in monetary units, then the individual value of the measurement to calculate a particular loss function, you will obtain on screen what it is shown in Figure 3.

Figure 3. Loss Function for limits and data provided from example 2.27 page 55 of book “Statistics”, Murray R. Spiegel, Mcgraw-Hill, 1961, 1988, 1997, 2002. New York. Obtained from software LOSSFUNCTION_EN, Note: The software uses n-1 into the formula to calculate the Standard Deviation and Variance, by hand calculation was made using n inside the formulas. That is why there is a little difference between calculates made by hand and those made with the software. Bibliography: Spiegel Murray R. “Statistics”, McGraw-Hill Interamericana Editores S.A. de C.V. México D.F. 2002. Taguchi Genichi “Introduction to Quality Engineering”, “Designing Quality into Products and Processes” Asian Productivity Organization, Tokyo 1986. Ross J. Phillip “Taguchi Techniques for Quality Engineering”, “Loss Function, Orthogonal Experiments, Parameter and Tolerance Design” Second Edition, McGraw-Hill Inc. New York, NY, 1988, 1996. “LOSSFUNCTION_EN” Software Carlos González González, México, 2007.

Carlos González González ASQ Fellow Master Black Belt ASQ Press Reviewer MBA National University San Diego CA USA e-mail: [email protected]