Resistor Production How can the outcomes be analysed to optimise the process? Resistor Production page: 1 of 15 Contents Initial Problem Statement 2 Narrative 3-11 Notes 12 Appendices 13-15
Resistor Production Initial Problem Statement Modern manufacturing plants make use of engineering quality control to ensure that a product s quality meets a specified standard and that rejection rates are minimised. When aiming to produce resistors to a specified value the process invariably produces a range of values. Often products are graded depending on how close they are to the nominal specification; those that are closer can have a higher sale price. How can the outcomes be analysed to optimise the process? Resistor Production page: 2 of 15
Narrative Introduction An electronics component plant manufactures electrical resistors. For such components all manufactured units are tested and graded according to their tolerance, i.e. how close the manufactured value is to the stated value. Those that are closest to the specified value can by sold at a higher price as being precision items. If you were in charge of the manufacturing process what would you measure and how? How would you use this information to grade the components? What else might the information you gather tell you? It is stated that every component is tested. Is this sensible? What is the alternative and what consequences does this have? Resistor Production page: 3 of 15
2. Measured data Looking at a particular machine responsible for manufacturing 100 Ω resistors the following data are collected for a production run of 10 000 components. The resistor values have been measured to the nearest whole number of Ohms. These data are plotted below Resistor value (Ω) Number of components <90 1 90 3 91 8 92 19 93 41 94 83 95 154 96 262 97 410 98 593 99 790 190 969 101 1096 102 1142 103 1096 104 969 105 790 106 593 107 410 108 262 109 154 110 83 >110 72 TOTAL 10 000 Resistor Production page: 4 of 15
Resistance values are recorded as a whole number but in fact the data are continuous so anything from 99.5 to 100.4999 is recorded as 100. Number 1200 1000 800 600 400 200 0 <90 90 91 92 93 94 95 96 97 98 99 100 101 102 103 Measured resistance recorded to nearest integer Figure 1. Does it matter as much whether the resistors produced have a resistance value of greater than 100 Ω as it does if they have a value of less than 100 Ω. What information about how the production process is working do these data tell you? 104 105 106 107 108 109 110 >110 Resistor Production page: 5 of 15
3. Calibrating the machine Look again at the production characteristics. Number 1200 1000 800 600 400 200 0 <90 90 91 92 93 94 95 96 97 98 99 100 101 102 103 Measured resistance recorded to nearest integer Figure 1. It is clear that the machine is not operating as intended. If you were recalibrating the machine what production characteristics would you address? Is it worth recalibrating the machine? 104 105 106 107 108 109 110 >110 Resistor Production page: 6 of 15
4. Grading the products For resistor production high precision is not usually required. However, they are not all sold as a single product. Instead the resistors are graded into 4 possible types depending on their tolerance (how close they are to 100 Ω) and sold at a price according to their quality. This allows a high yield from the machine without requiring a high precision process. The tolerance bands are summarised below. In this case, when assigning resistors to the bands the recorded values of the resistance are used which are to the nearest whole number. Tolerance Description Unit price Within 1% Premium quality product 0.016 Within 2% High quality product 0.012 Within 5% Standard product 0.005 Within 10% Low quality product 0.001 Outside 10% Reject - These unit prices are given to one-thousandth of a pound. Is this possible? What does it mean and why would a company do this? Resistor Production page: 7 of 15
Activity 1 Look again at the production data. Resistor value (Ω) Number of components <90 1 90 3 91 8 92 19 93 41 94 83 95 154 96 262 97 410 98 593 99 790 190 969 101 1096 102 1142 103 1096 104 969 105 790 106 593 107 410 108 262 109 154 110 83 >110 72 TOTAL 10 000 What are the yields for each of the categories of product? Remember that each product can belong to only one category. Use the information to fill in the following table. Per 10 000 produced Tolerance Number Sales value Within 1% Within 2% Within 5% Within 10% Outside 10% Total Look at the total row in the table. What do the totals tell you? How shall you decide whether or not to recalibrate the machine? Resistor Production page: 8 of 15
5. Predicted behaviour of the calibrated machine For a nominal 100 Ω resistor the expected distribution of resistance values when measured to the nearest whole value per 10,000 components manufactured are summarised in the table below. Resistor value (Ω) Number of components <90 12 90 19 91 41 92 83 93 154 94 262 95 410 96 593 97 790 98 969 99 1096 190 1142 101 1096 102 969 103 790 104 593 105 410 106 262 107 154 108 83 109 41 110 19 >110 12 Resistor Production page: 9 of 15
These data are plotted below. Number 1200 1000 800 600 400 200 0 <90 90 91 92 93 94 95 96 97 98 99 100 101 102 103 Measured resistance recorded to nearest integer Figure 2. Is this distribution reasonable? 104 105 106 107 108 109 110 >110 Resistor Production page: 10 of 15
These data are plotted below. Activity 2 What are the yields for each of the categories of product in the recalibrated machine? Remember that each product can belong to only one category. Use the information to fill in the following table and compare the values with the previous results Per 10 000 produced Tolerance Number Sales value Within 1% Within 2% Within 5% Within 10% Outside 10% Total The results for the measurements made prior to any calibration are Per 10 000 produced Tolerance Number Sales value Within 1% 2855 45.680 Within 2% 1735 20.820 Within 5% 3681 18.405 Within 10% 1656 1.656 Outside 10% 73 0.000 Multimedia Total 10 000 86.561 What are the rejection rates for the current and predicted recalibrated operations? Give the answers in terms of a decimal, a fraction and a percentage. What is the increase in sales income as a percentage? Would you fix the machine? The resource Resistor Production Interactive is available to show how variations of mean and spread affect the results. See appendix 1. Resistor Production page: 11 of 15
Notes Resistor production numbers At the time of writing the Sichuan Yongxing Electronics Co. based in China has the capacity to produce 5 billion resistors per year! These will be manufactured in a range of values and product types (high power, lower power, variable, etc) but the number still represents an enormous production quantity for any individual item. The range of values produced is not random. It is designed to give the best possible range of products for use. The determination of the best spread of such standard values is covered in a separate resource. Resistor Production page: 12 of 15
Appendix 1 using the interactives Resistor Production Interactive These are some guidelines for using the resource Resistor Production Interactive. Figure 3. Use the sliders to change the mean and standard deviation. Pressing the random button will select random values for these parameters. Pressing the show button at the bottom of the page will display the distribution. This allows you to guess the shape before viewing it. Resistor Production page: 13 of 15 Figure 4.
Pressing the show button at the top of the page when a new distribution has been selected and shown (using the bottom show button will reveal the value per 10 000 items for the selected distribution. Try to guess whether it will be larger or smaller than the previous case or the calibrated (nominal) case. Figure 5. If the live update box is ticked the distribution and value will change as the parameters are varied. Resistor Production page: 14 of 15
Appendix 2 mathematical coverage PL objectives Use statistics to solve engineering problems Display data using a bar chart Extract numerical information from a data set Calculate probability Estimate probability as a relative frequency Resistor Production page: 15 of 15