Loss Function Cgg2

Author: Carlos González González Article: Taguchi Loss Function Condition: Higher 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.

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Figure 1. Goalpost Philosophy inside it is good outside it is bad Figure 2 shows that the exponential loss function increase its value when the value of the items are closer to the minimum value permitted, showing too that the minimum cost stays on the maximum values.

Figure 2. Exponential Quality Loss Function (Higher is Better) and Probability Density Function (Normal Curve – Bell Shape). PhD. Genichi Taguchi uses the Mathematical equation for higher is better (1-i): L = k[1 / y^2 ]

(1-i)

In this equation, L is the loss associated with a particular value y, k is a constant depending on the cost at the lower limit of specification. Applying this equation to the next problem we will have: “The following table shows the resistance to rupture in Pascals of 125 samples with a Lower Specification Limit LSL = 7.75 at this limit the Loss in monetary units is L = $1.00, a desired possible reading of 11.0 on the upper side, and the inspector want to know how much is the loss of a piece with a measurement of 10.5 Pascals. Table 1-i 8.9 8.6 8.8 8.9 8.6 9.4 9.1 9.7 8.9 8.3 8.9 8.5 8.3 8.4 9.3 8.5 8.6 9.1 9.3 8.8 8.5 9.0 8.3 8.7 8.3 8.3 8.9 9.1 9.0 9.0 9.0 9.3 8.9 8.8 9.2 8.8 8.7 8.6 8.8 8.8 9.1 8.6 9.2 8.6 8.7 8.8 8.7 8.6 9.7 8.6 9.2 8.4 9.1 9.1 8.8 8.8 8.9 8.7 8.8 8.6 8.7 8.4 8.8 8.9 9.1 9.2 9.1 8.8 9.0 9.2 8.8 9.7 8.3 9.3 9.3 8.5 8.9 8.2 8.9 8.7 8.5 8.4 8.0 8.7 8.8 8.8 8.7 9.0 8.7 8.4 8.6 9.0 8.5 8.7 9.6 9.0 8.8 9.1 8.4 9.1 8.5 8.8 8.5 8.3 8.7 8.7 9.1 9.0 9.0 8.5 8.9 8.2 9.0 8.7 8.9 9.4 9.2 8.7 8.7 8.9 9.2 8.8 8.5 8.6 9.7

Substituting values on equation (1-i) we have: $1.00 = k(1/ y^2) The lower specification limit (LSL) is substituted into the equation, which is where the $1.00 loss is incurred. Solving for k, k = $1.00 (y^2) Giving that y = 7.75, k = $1.00 (7.75)^2 k = $1.00 (60.0625) k = $60.0625 per (pascal)^2 Therefore for a part with value of 10.50: L = 60.0625[1 / (y^2)]

(1-ii)

L = 60.025 [ 1 / (10.50)^2] L = 60.025 (1/ 110.25) L = 60.025 (0.00907029) = $0.5444 A method of estimating average loss per part entails using the loss equation with a different form. For a large number of parts, the average loss per part is equal to. L = k[1/(ybar)^2] [1 + {(3S^2)/(ybar)^2}]

(1-iii)

S^2 = variance around the average, ybar ybar = average value of y for the group. k = constant of loss For the set of values presented in Table 1-i the values of S^2 and ybar and k can be calculated by hand. S^2 = (0.3363467)^2 pascal^2 = 0.113129

ybar = 8.824 Using Equation 1-iii we have: L = k[1/(ybar)^2] [1 + {(3S^2)/(ybar)^2}] L = 60.0625[1 / (8.824^2)] [1 +{(3x0.113129)/(8.824^2)}] L = 60.0625[1 / (77.8629)] [1 +{(0.33939)/(77.8629)}] L = 60.0625(0.012843) (1 + 0.0043588) L = (0.7709) (1.0043588) = $ 0.77426 L = $0.77426 per part We were working for an objective named “higher is better”. Suppose that these 125 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 0.77426 = $ 3871.30 Because they are not produced as higher as possible 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 (125 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 [HIGHER IS BETTER] then you should provide the Lower Limit of Specification, The highest possible measurement 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 (125 measurements). 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 calculations 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]