Loss Function Cgg3

Author: Carlos González González Article: Taguchi Loss Function Condition: Lower 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 exponential loss function increase its value when the value of the items are closer to the maximum value permitted, showing too that the minimum cost stays on the minimum values.

Figure 2. Exponential Quality Loss Function (Lower is Better) and Probability Density Function (Normal Curve – Bell Shape). PhD. Genichi Taguchi uses the Mathematical equation for lower is better (1-i): L = k[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 upper specification limit. Applying this equation to the next problem we will have: Problem 4.37, Chapter 4, page 110 book “Statistics” Authors Murray R. Spiegel and Larry J. Stephens, Edit. McGraw-Hill Interamericana Editores S.A. de C.V. México, 2002. “The following table shows the number of weeks to get a job to 25 men after they were fired from a company, the minimum amount of weeks that it is possible to get a job is 2 weeks. The Upper Specification Limit USL = 26, at this limit the Loss in monetary units is L = $2000, we want to know how much is the loss of a man that got a job after 15 weeks. Table 1-i 13 22 13 26 17 17 7 13 22 14

16 7 6 18 20

10 17 11 10 15

16 8 16 21 11

Substituting values on equation (1-i) we have: $2000 = k(y^2)

The Upper Specification Limit (USL) is substituted into the equation, which is where the $2000 loss is incurred. Solving for k, k = $2,000 / (y^2) Giving that y = 26, k = $2,000 / (26)^2 k = $2,000 / (676) k = $2.9585 per (week)^2 Therefore for a man that got the job after 15 weeks: L = 2.9585[(y^2)]

(1-ii)

L = 2.9585 [ (15)^2] L = 2.9585 (225) L = $665.66 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[(S^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 = (5.1137)^2 week^2 = 26.1504 ybar = 14.64

k = 2.9585 Using Equation 1-iii we have: L = k[(S^2) + (ybar)^2] L = 2.9585 [(5.1137^2) + (14.64)^2] L = 2.9585[(26.1504) + (214.3296)] L = 2.9585[240.48] L = $ 711.46 per man We were working for an objective named “lower is better”. Suppose that these 25 men were drawn from a lot of 5,000 men the total loss will be the multiplication of the average loss per man times the total number of men in the lot. Total Loss = 5,000 x $711.46 = $ 3,557,300 Because the men can not obtain a job as soon 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 (25 data) 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 [LOWER IS BETTER] then you should provide the desired minimum possible, The Upper Specification Limit 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 (25 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 it 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]