many quality problems remain invisible to consumers

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1 Quality Management 1. Define Quality and TQM. 2. What are the ISO standards and why are they important? 3. What is Six Sigma? 4. Explain how benchmarking is used? 5. What are quality robust products and Taguchi concepts? 6. What are the seven tools of TQM? 7. Explain why control charts are used. 8. Explain the use of x-chart and R charts. 9. List the five steps used in building control charts. 10. When are p and c charts used? 11. Explain C p and C pk 122 / 469 Quality Management Many production and service process suffer from quality problems. airlines lose baggages computer manufacturers ship laptops with defective disk drives pharmacies distribute wrong medications to patients postal services lose or misdeliver articles by mail In addition to these quality problems directly visible to consumer, many quality problems remain invisible to consumers. Although hidden to the consumer, such quality problems have a profound impact on the economics of business processes. 123 / 469

2 Defining Quality Several Definitions: Quality refers to the ability of a good or service to consistently meet or exceed customer expectations.... the quality of a product or service is the degree to which the product or service meets specifications. Render, Stair and Hanna Quality is the degree to which a specific product conforms to a design or specification. H.L Gilmore Quality is the totality of features and characteristics of a product or service that bears on its ability to satisfy stated or implied needs. R. Johnson and O William Quality is fitness to use. J.M. Juran 124 / 469 Quality or Lack of Quality, Impacts the Entire Organization 125 / 469

3 Key Dimensions of Quality Performance-technical capability Durability- the lifespan of product before it begins to deteriorate Special Features- features that are merely nice to have Serviceability-support after the initial sell of the product Reliability- the likelihood of a product working properly Aesthetics- psychological impressions Conformance-the consistency with which the firm meets the specifications Perceived quality Safety Value = quality cost 126 / 469 Costs of Quality Prevention costs - are associated with reducing/avoiding the potential for defects before they happen. Appraisal costs - are associated with evaluating (i.e., assess and inspect) products, parts, and services. Internal failure - costs resulting from defects that are discovered during the production of a product or service. External costs - arise when a defect is discovered after the customer has received the product or service. 127 / 469

4 Costs of Quality: In Practice Toyota recalls CBC: August 26, 2010: Toyota recalls 1.13 million Corolla, Matrix cars; affects 136,000 Corollas, 64,300 Matrixes in Canada CBC: January 27, 2011 Toyota recalls 1.7 million vehicles worldwide Toyota shares fell 28 cents to $68.78 US. Antenna problems (2010): iphone 4 recall would cost $1.5 billion Apple gave rubber cases to iphone 4 customers - retail price of $29 per unit Boeing halts 787 deliveries until batteries fixed Maple Leaf Foods listeriosis outbreak (CBC, ) Twenty-three (23) people died and there were 57 total confirmed cases. About 250 employees were laid off while the plant was being cleaned The recall reportedly cost the company $20 million. Four separate class-action lawsuits were filed in Ontario, Quebec, Saskatchewan, and British Columbia. The lawsuit in Ontario is claiming damages of $350 million. The lawsuits were settled in December 2008 for $27 million. 128 / 469 Total Quality Management (TQM) TQM is a philosophy that stresses three principles: customer satisfaction employee involvement, and continuous improvement in quality. Seven Concepts of TQM Continuous improvement Six Sigma Employee empowerment Benchmarking Just-in-time (JIT) Taguchi concepts Knowledge of TQM tools 129 / 469

5 Six Sigma and Three Sigma Six Sigma is a technique created by Motorola Inc. to manage process variations that causes defects 6 ; and to systematically work towards managing variation to eliminate those defects. Two meanings of Six Sigma 1) Statistical definition of a process that is % capable, 3.4 defects per million opportunities (DPMO) 2) A comprehensive system for achieving and sustaining business success. Six Sigma is a program designed to reduce defects, lower costs, and improve customer satisfaction 6 Defects are defined as unacceptable deviation from the mean or target. 130 / 469 Six Sigma 131 / 469

6 Why 99 % Quality Level Is Not Acceptable? At major airports, 99 % quality means two unsafe plane landings per day. In mail processing, 99 % quality means 16,000 pieces of lost mail every hour. In power generation, 99 % quality will result in 7 hours of no electricity each month. In medical surgery, 99 % quality means 500 incorrect surgical operations per week. 132 / 469 Try It Out: Six Sigma Program If 2.5 million passengers pass through the Pearson International Airport with checked baggage each month, a successful Six Sigma program for baggage handling would result in how many passengers with misplaced luggage? (a) 3.4 (b) 6750 (c) 8.5 (d) 2700 (e) 6 times the monthly standard deviation of passengers (f) none of the above. 133 / 469

7 Try It Out: Three Sigma Program If 2.5 million passengers pass through the Pearson International Airport with checked baggage each month, a successful Three Sigma program for baggage handling would result in how many passengers with misplaced luggage? (a) 3.4 (b) 6750 (c) 8.5 (d) 2700 (e) 6 times the monthly standard deviation of passengers (f) none of the above. 134 / 469 Seven Tools of TQM Tools for Generating Ideas Check sheets Scatter diagrams Cause-and-effect diagrams Tools to Organize the Data Pareto charts Flowcharts Tools for Identifying Problems Histogram Statistical process control chart 135 / 469

8 Check Sheet Check list is a form used to record the frequency of occurrence of certain product or service characteristics related to quality. 136 / 469 Scatter Diagram A scatter diagram is a plot of two variables showing whether they are related. 137 / 469

9 Cause-and-Effect Diagram A Cause-and-Effect Diagram is a diagram that relates key quality problem to its potential causes. 138 / 469 Pareto Chart A Pareto chart is a bar chart on which factors are plotted in decreasing order of frequency along the horizontal line. 139 / 469

10 Flowchart (Process Diagram) A flowchart is a diagram that presents a process or system using annotated boxes and interconnected lines. 140 / 469 Histogram A histogram is a summarization of data measured on a continuous scale, showing the frequency distribution of some quality characteristic (in statistical terms, the central tendency and dispersion of the data). 141 / 469

11 Statistical Process Control Chart A statistical process control is the application of statistical techniques to determine whether the output of a process conforms to the product or service design. SPC control charts are graphic presentations of data over time that show upper and lower limits for the process we want to control. The control charts are constructed in such as that new data can be quickly compared with past performance data. 142 / 469 Statistical Quality Control The best companies design quality into the process (e.g., by undertaking continuous improvement and Six Sigma quality projects), thereby greatly reduce he need for inspections 7 /tests. The least progressive companies rely heavenly on receiving and shipping inspections/tests. Many occupy a middle ground. Statistical quality control uses statistical techniques and sampling in monitoring and testing of quality of goods and services. 7 Inspection is an appraisal activity that compares the quality of a good or service to a standard. 143 / 469

12 Statistical Process 8 Control The objective of a process control system is to provide a statistical signal when assignable causes of variation are present; and help detect and eliminate assignable causes of variation. If the sample results fall outside the limits, the operator stops the process and takes corrective measures. SPC is concerned with statistical evaluation of the product in the production process. To do SPC, the operator takes periodic samples from the process and compares them with predetermined limits. If the sample results fall within the limits, the operator allows the process to continue. 8 A process is a group of related tasks with specific inputs and outputs. Process exists to create value for the customer, the shareholder, or society. 144 / 469 TheTwoTypesofVariations Process variation is the root cause of all quality problems. The two types of Variations: Natural/Chance or common variation: refers to constant variation reflecting pure randomness in the process. Special or assignable variation: is a non-random variability in process output, a variation whose cause can be identified (e.g., excessive tool wear, equipment that needs adjustment, defective materials, and human errors). 145 / 469

13 Controlling Variations The main task in quality control is to distinguish assignable from random variation. How? Complete inspection: used when the costs of passing defects to the next workstation or to external customers outweigh the inspection costs. Sampling: a well-planned sampling plan can approach the same degree of protection as complete inspection. 146 / 469 Samples To measure the process, we take samples and analyze the sample statistics following these steps 1. Samples of the product, say five boxes of cereal taken off the filling machine line, vary from each other in weight 2. After enough samples are taken from a stable process, they form a pattern called a distribution 3. There are many types of distributions, including the normal (bell-shaped) distribution, but distributions do differ in terms of central tendency (mean), standard deviation or variance, and shape 4. If only natural causes of variation are present, the output of a process forms a distribution that is stable over time and is predictable 5. If assignable causes are present, the process output is not stable over time and is not predicable 147 / 469

14 Sample and Sample Distribution 148 / 469 Process Control 149 / 469

15 Quality Measurement - Types of Data Variables: product or service characteristics, such as weight, length, volume, or time, that can be measured. Characteristics that can take any real value May be in whole or in fractional numbers Continuous random variables Attributes: product or service characteristics that can quickly counted for acceptable quality (i.e., yes-or-no) Defect-related characteristics Classify products as either good or bad or count defects Categorical or discrete random variables 150 / 469 Central Limit Theorem (CLT) The CLT is the theoretical foundation of x-charts, states that regardless of the distribution of the population of all parts or services, the distribution of sample means, x s (each of which is a mean of a sample drawn from the population), will tend to follow a normal curve as the sample size grows larger. Even if n is fairly samll (say 4 or 5), the distributions of the averages will still roughly follow a normal curve. The CLT also states that: The mean of the sampling distribution ( x) will be the same as the population mean μ. x = μ The standard deviation of the sampling distribution (σ x ) will equal the population standard deviation (σ) divided by the square root of the sample size, n σ x = σ n 151 / 469

16 The Relationship Between Population and Sampling Distributions Because this is a normal distribution, we can state that % of the time, the sample averages will fall within ±3σ x of the population if the process has only random variations % of the time, the sample averages will fall within ±2σ x of the population if the process has only random variations. 152 / 469 Control Charts for Variables A control chart is a time-ordered plot of a sample statistic, with limits. Control charts for variables consist of two parts: Sample mean control chart - x-charts Range control chart Two approach to sample mean chart x-charts when population (process) standard deviation, σ, isknown x-charts when population (process) standard deviation, σ, is unknown 153 / 469

17 x-charts When Population (Process) Standard Deviation, σ, isknown Control limits for x-charts Upper control limit(ucl x )= x + zσ x where: Lower control limit(lcl x )= x zσ x σ x = σ x n = standard deviation of the sampling distribution of the mean x = mean of the sample means; z = number of normal standard deviations (e.g., 2 for 95.45% confidence, 3 for 99.73%) σ x =the population standard deviation. 154 / 469 Figure It Out: Quality Inspection A quality inspector took five samples, each with four observations,of the length of a part (in cm). The inspector knows from previous experience that the standard deviation, σ, of the process is 0.02 cm. Sample Observation Obtain Three Sigma (i.e., z=3) control limits for sample mean. Is the process in control? 2. If a new sample has values 12.10, 12.19, 12.12, and 12.14, using the sample mean control chart in (1), has the process mean changed (i.e., is there an assignable variation)? 155 / 469

18 Solution: Obtain Three Sigma (i.e., z=3) Control Limits for Sample Mean. Is The Process In Control? sample Observation x x = =12.11cm 5 z=3; σ =0.02, n=4. σ x = σ = 0.02 =0.02/2 =0.01 n 4 UCL x = = 12.14cm UCL x = = 12.08cm Because all the five sample means fall between the control limits (12.08, 12.14), the process is in control. 156 / 469 Sample Mean Control Limits 157 / 469

19 Sample Mean Control Limits Because all the five sample means fall between the control limits (12.08, 12.14), the process is in control. 158 / 469 Solution If a new sample has values 12.10, 12.19, 12.12, and 12.14, using the sample mean control chart in (1), has the process mean changed (i.e., is there an assignable variation)? sample Observation x x 6 = = cm 4 Because < < 12.14, the process mean has not changed (i.e., there is only random variation). Note: Because individual values are represented by the process distribution, the fact that some individual measurements fall outside of the control limits is irrelevant. 159 / 469

20 x Chart When Population (Process) Standard Deviation, σ, is Unknown Control limits for x-charts where: Upper control limit(ucl x )= x + A 2 R Lower control limit(lcl x )= x A 2 R R = average range of the samples; A 2 = a value from Table S6.1 (p. 233); and x = mean of the sample means. 160 / 469 x Chart When Population (Process) Standard Deviation, σ, is Unknown sample Observation x R R A 2 =0.729 (Table S6.1) for n =4, R =0.046, x =12.11cm UCL x = = 12.14cm LCL x = = 12.08cm The process is in control. 161 / 469

21 Factors for Calculating Control Chart Limits- Three Sigma 162 / 469 x Chart When Population (Process) Standard Deviation, σ, is Unknown Twenty sample of n = 8 have been taken of the weight of a part. The average of sample ranges for the 20 samples is kg, and the average of sample means is 3 kg. The Three Sigma control limits for sample means of this process is: 1. UCL x =3.006, LCL x = UCL x =0.370, LCL x = UCL x =3.006, LCL x = UCL x =2.994, LCL x = UCL x =2.994, LCL x = none of the above. 163 / 469

22 x Chart When Population (Process) Standard Deviation, σ, is Unknown Twenty sample of n = 8 have been taken of the weight of a part. The average of sample ranges for the 20 samples is kg, and the average of sample means is 3 kg. The Three Sigma control limits for sample means of this process is (to three decimal places): 1. UCL x =3.006, LCL x = UCL x =0.370, LCL x = UCL x =3.006, LCL x = UCL x =2.994, LCL x = UCL x =2.994, LCL x = none of the above. A 2 =0.373 (Table S6.1) for n =8, R =0.016, x =3kg UCL x = = 3.006kg LCL x = = 2.994kg 164 / 469 Sample Range Control Chart - R-Chart Sample range 9 control chart (R-chart) is used to monitor process dispersion or spread. Control limits for the sample range control charts are found using the average sample ranges from the following formula: UCL R = D 4 R LCL R = D 3 R where the D 3 and D 4 values are obtained from Table S A range is the difference between the smallest and the largest values in a sample 165 / 469

23 Try It Out Suppose that a large production lot of boxes of cornflakes is sampled every hour. To set control limits that included 99.7% of the sample means, 36 boxes are randomly selected and weighted. The standard deviation of the overall population of boxes is estimated to be 2 ounces. The average mean of all samples taken is 10 ounces. Determine the control limits. 2. Consider a process in which the average range is 53 pounds. If the sample size is 5, we want to determine the upper and lower control charts. 10 ANS: 1) UCL x = 9 ounces;lcl x = 11 ounces 2) UCL x = pounds;lcl x =0 pounds 166 / 469 R Chart: Calculate Range (R) sample Observation x R R R 1 = max{12.11, 12.10, 12.11, 12.08} min{12.11, 12.10, 12.11, 12.08}=0.03 R 2 = max{12.15, 12.12, 12.10, 12.11} min{12.15, 12.12, 12.10, 12.11}=0.05 R 3 = max{12.09, 12.09, 12.11, 12.15} min{12.09, 12.09, 12.11, 12.15}=0.06 R 4 = max{12.12, 12.10, 12.08, 12.10} min{12.12, 12.10, 12.08, 12.10}=0.04 R 5 = max{12.09, 12.14, 12.13, 12.12} min{12.09, 12.14, 12.13, 12.12}= / 469

24 R chart : R, UCL R, UCL L R = =0.046 D 3 =0.000 and D 4 =2.282 (Table S6.1) for n =4, R =0.046cm UCL R = D 4 R = = 0.105cm The process range is in control. LCL R = D 3 R = = 0cm 168 / 469 Figure It Out: R-chart Twenty-five sample of n = 10 observation have been taken from a milling process. The average of sample ranges was 0.01 centimeter. The upper and lower control limits for sample range of this process is (to three decimal places): 1. UCL x =0.01, LCL x = UCL x =0.018, LCL x = UCL x =0.223, LCL x = UCL x =0.002, LCL x = UCL x =1.777, LCL x = none of the above. 169 / 469

25 Solution: R-chart Twenty-five sample of n = 10 observation have been taken from a milling process. The average of sample ranges was 0.01 centimeter. The upper and lower control limits for sample range of this process is (to three decimal places): 1. UCL x =0.01, LCL x = UCL x =0.018, LCL x = UCL x =0.223, LCL x = UCL x =0.002, LCL x = UCL x =1.777, LCL x = none of the above. D 3 =0.223 and D 4 =1.777 from (Table S6.1) for n = 10, R =0.01cm UCL R = D 4 R = = 0.018cm LCL R = D 3 R = = 0.002cm 170 / 469 Solution: R-chart D 3 =0.223 and D 4 =1.777 from (Table S6.1) for n = 10, R =0.01cm UCL R = D 4 R = = 0.018cm LCL R = D 3 R = = 0.002cm Interpretation of the above results: A sample range of cm or more would suggest that the process variability had increased; suggesting that the process is producing too much variation; we would want to investigate this in order to remove the cause of variation. A sample range of cm or less would imply that the process variability had decreased; suggesting that even though a decreased variability is desirable, we would want to determine what is causing it; perhaps an improved method has been used, in which case we would want to identify it. 171 / 469

26 Can We Find σ from R? If the standard deviation of the process is unknown but the average of sample ranges R is known, then we can find σ by equating the UCL x for known σ to the UCL x for unknown sigma (assuming the data are normally distributed): x + zσ x = x + A 2 R zσ x = A 2 R z σ n = A 2 R σ n z A 2 R 172 / 469 Mean and Range Charts Sample mean and range control charts provide different perspectives on a process. Sample mean control charts are sensitive to shifts in the process mean, whereas sample range control charts are sensitive to changes in the process dispersion. 173 / 469

27 Mean and Range Charts The use of both charts provides complete information than either chart alone; by using both charts we can track changes in the process distribution. 174 / 469 Five Steps to Follow in Using x and R-Charts 1. Take samples from the population and compute the appropriate sample statistic. 2. Use the sample statistic to calculate control limits and draw the control chart. 3. Plot sample results on the control chart and determine the state of the process (in or out of control). 4. Investigate possible assignable causes and take any indicated actions. 5. Continue sampling from the process and reset the control limits when necessary. 175 / 469

28 Control Charts for Attributes Control charts for attributes are used when the process characteristic is counted rather than measured, i.e., an item is either defective or not, good or bad, yes or no, acceptable or unacceptable. There are two types of attribute control charts: 1. p-chart: for the fraction (percent) of defective items in a sample 2. c-chart: for the number of defects per unit 176 / 469 Control Limits for p-charts The theoretical basis for p-chart is the binomial distribution, but applying the Central Limit Theorem allows us to assume a normal distribution for the sample statistics. A p-chart is used to monitor the proportion of defective items generated by a process. 1. The centre line on a p-chart is the average fraction defective in the population, p. 2. The standard deviation of the sampling distribution of sample proportion, when p is known p(1 p) σ p = n 3. Control limits are calculated using the formulas (14) UCL p = p + zσ p LCL p = p zσ p If p is unknown, it can be calculated from samples. That estimate, p, replaces p in the above formulas. Because the formula is an approximation, sometimes the calculated LCL p is negative. In this case, zero is used as the lower limit. 177 / 469

29 Figure It Out: Control Limits for p-charts An inspector counted the number of defective parts a prototype machine made over 20 hours, taking a sample of 100 every hour. Sample Number of Defectives Sample Number of Defectives Construct a three-sigma control chart for the sample proportion of defectives. Is the machine producing a stable proportion of defectives? 178 / 469 Solution: Control Limits for p-charts n = 100; total number of defective = 220; p = total number of defectives total number of observations = (100) =0.11 (15) p(1 p) 0.11(1 0.11) σ p = = n 100 Control limits are calculated using the formulas =0.03 (16) UCL p = p + z ˆσ p = = 0.20 LCL p = p z ˆσ p == = / 469

30 Number and Proportion of Defectives Sample Defectives Proportion Sample Defectives Proportion / 469 Plotting the Control Limits The process is not in control: sample 8 (=22/100 = 0.22) and sample 15 (21/100=0.21) are above the upper control limit. Therefore, the machine operation should be investigated fpr assignable causes, and after corrective actions, new data should be collected and new p-control limits should be calculated. 181 / 469

31 Control Limits for c-charts When the goal is to control the number of occurrences (e.g., defects) per unit, a c-chart is used. In this case, there is no sample size, only occurrences may be counted; the nonoccurrences cannot be. The theoretical basis for c-chart is a Poisson distribution, but applying the Central Limit Theorem allows us to assume a normal distribution for the sample statistics. The mean number of defects per unit is c and the standard deviation is (c). For practical reasons, the Normal approximation to the Poisson is used. The control limits are calculated using the formulas UCL c = c + z c LCL c = c z c If the value of c is unknown, the sample estimate, c, isusedinplaceof c, where number of defects c = number of samples When the computed lower limit is negative, it is set to zero. 182 / 469 Figure It Out Rolls of coiled wire are monitored using a c-chart. Eighteen rolls have been examined, and the number of defects per roll has been recorded in the following table. Is the process in control? Sample Defectives Sample Defectives / 469

32 Solution Total number of defects = 45; total number of samples =18. c = number of defects number of samples =45/18 = 2.5 UCL c = c + z c = =7.24 LCL c = c z c = = The process is in control. Suppose a new roll has seven defects. Using the above control limits, is the process producing the same average number of defects? The answer is yes, because 0 < 7 < 7.24, the process is still producing a stable number of defects. 184 / 469 Managerial Issues and Control Charts Three major management decisions: Select points in the processes that need SPC Which parts of the job are critical to success? Which parts of the job have a tendency to become out of control? Determine the appropriate charting technique Set clear policies and procedures 1. Should the assembly line be stopped if the average length of five successive samples is above the centerline? 185 / 469

33 Patterns to Look for on Control Charts 186 / 469 Design Specifications and Process Capability So far, we focussed on the question to what extent the process is in control. However, it is important to understand that a process that is in control might still fail to deliver the quality demanded from the customer or a downstream operation in the process. The natural variation of a process should be small enough to produce products that meet the standards required A process in statistical control does not necessarily meet the design specifications Process capability is a measure of the relationship between the natural variation of the process and the design specifications The consistency requirement from the customer typically takes the form of a design specification. A design specification includes: atargetvalue a tolerance, describing the range of values that are acceptable from the customer s perspective. Two Popular Measure of Process Capability Process Capability Ratio, C p Process Capability Index, C pk. 187 / 469

34 Process Capability Ratio, C p The process capability ratio, C p,iscalculatedas: C p = Upper Specification Lower specification 6σ (17) For a process to be capable, its values must fall within upper and lower specifications. This typically means the process capability is within ±3 standard deviations, σ, from the process mean. Since this range of values is 6 standard deviations, 6σ, a capable process tolerance, which is the difference between the upper and lower specifications, must be greater than or equal to 1. A capable process must have a C p of at least 1.0 Does not look at how well the process is centered in the specification range Often a target value of C p =1.33isusedtoallowforoff-center processes Six Sigma quality requires a C p = / 469 Process Capability Index Process capability index C pk,iscalculatedas: [ USL X C pk = minimum of, 3σ where USL = upper specification limit LSL = lower specification limit C pk = process capability index ] X LSL 3σ (18) 189 / 469

35 Figure It Out: Process Capability Ratio, C p The specifications for a manifold gasket that installs between two engine parts calls for a thickness of 2.500mm ± 0.020mm. The standard deviation of the process is estimated to be mm. 1. What are the upper and lower specification limits for this product? 2. The process is currently operating at a mean thickness of 2.50 mm. What is the C p for this process? 3. The purchaser of these parts requires a capability index of Is this process capable? Is this process good enough for the supplier? 4. If the process mean were to drift from its setting of mm to a new mean of 2.497, would the process still be good enough for the supplier s needs? 190 / 469 Solution: Process Capability Ratio, C p 1. What are the upper and lower specification limits for this product? LSL = 2.48 mm; USL = 2.52 mm. 2. What is the C p for this process? C p = = The purchaser of these parts requires a capability index of Is this process capable? Is this process good enough for the supplier? Yes to both parts of the question. 4. If the process mean were to drift from its setting of mm to a new mean of mm, would the process still be good enough for the supplier s needs? The C pk index is now relevant, and its value is the lesser of and C pk = minimum of [ ] , C pk = minimum of (1.917, 1.417) = The process is still capable, but not to the supplier s needs. 191 / 469

36 Try It Out: Process Capability Ratio, C p 11 The specification for a plastic handle calls for a length of 6.0 inches ± 0.2 inches. The standard deviation of the process is estimated to be 0.05 inches. 1. What are the upper and lower specification limits for this product? 2. The process is known to operate at a mean thickness of 6.1 inches. What is the C p and C pk for this process? Is this process capable of producing the desired part? 11 ANS: LSL = 5.8 inches, USL = 6.2 inches. Cp = Cpk = / 469 Meaning of C pk Measures 193 / 469

37 Reliability- Chapter 17 Reliability is a dimension of quality related to product design, and refers to the probability of a product will be functional when used. Systems are composed of a series of individual interrelated components, each performing a specific job. In any one component fails to perform, for whatever reason, the overall system can fail. The reliability of a complete product equals the product of all the reliabilities of the modules, that is: where R s =(R 1 )(R 2 )..(R i )(R n ) (19) Rs = reliability of the complete product n = the number of modules, components, or subsystems Ri = reliability of each module, component or subsystem i. This measure assumes that the reliability of each component or subsystem is independent of the other. 194 / 469 Figure It Out The reliability of the three components of a Portable joggers radio each will still be operating two years from now. a motherboard with a reliability of 0.99 a housing assembly with a reliability of 0.90 a headphone set with a reliability of 0.85 The reliability of the portable radio is: R s =(0.99)(0.90)(0.85) = 0.76 Suppose that new designs resulted in a reliability of 0.95 for the housing and 0.90 for the headsets. Product reliability would improve to: R s =(0.99)(0.95)(0.90) = / 469

38 Product Reliability and the Number of Components Overall system reliability as a function of the number of n components - each with the same reliability- and component reliability with components in a series. 196 / 469 Providing Redundancy - Using Components in Parallel The technique here is to backup components with additional components. Redundancy is provided to ensure that if one component fails,the system has resource to another. For example 1, suppose the reliability of a component is 0.9 and we back it up with another component with reliability of 0.9. The resulting reliability is the probability of the first component working plus the probability of the backup (or parallel) component working multiplied by the probability of needing the back up component (i.e., =0.1) RS =( (1 0.9)) = / 469

39 Parallel System Reliability The reliability of a parallel system is given by: R s =1 (1 R 1 )(1 R 2 )..(1 R i )(1 R n ) (20) where R s = reliability of the complete product n = the number of modules, components, or subsystems R i = reliability of each module, component or subsystem i. (1 R i ) = the probability of needing the back up component Example: suppose the reliability of a component is 0.9 and we back it up with another component with reliability of 0.8. The resulting reliability is the probability of the first component working plus the probability of the backup (or parallel) component working multiplied by the probability of needing the back up component (i.e., =0.1) RS =( (1 0.9)) = 0.98 RS =(1 (1 0.9) (1 0.8)) = / 469 Figure It Out Which product design, A or B, has higher reliability? B is designed with back up units for components, R3 and R ANS: Design A s Rs = ; Design B s Rs = / 469

40 Solution Design A: R s =(R 1 ) (R 2 ) (R 3 ) (R 4 )= = Design B: R s =(R 1 ) (R 2 ) (1 (1 R 3 )(1 R 3 )) (1 (1 R 4 )(1 R 4 )) R s = (1 ( )(1 0.95)) (1 (1 0.99)(1 0.99)) = (0.9993) (0.9999) = / 469 Try It Out In a series system, adding an additional part to a component or product ordinarily reduces reliability by introducing an additional source of failure. (True or False) 2. A redundant part or component increases reliability because it is connected in parallel, not in series. (True or False) 3. What is the reliability of a four-component product, with components in series, and component reliabilities of.90,.95,.98, and.99? (A) under 0.83 (B) 0.90 (C) (D) no less than.99 (E) none of the above 4. A system is composed of three components A, B, and C. All three must function for the system to function. There are currently no backups in place. The system has a reliability of If a backup is installed for component A, the new system reliability will be (A) unchanged (B) less than (C) greater than it would be if a backup were also installed for component B (D) greater than (E) none of the above 13 ANS: 1) T 2) T 3) A 4) D 201 / 469

41 Figure It Out A quality analyst wants to construct a sample mean chart for controlling a packaging process. The analyst knows from past experience that the process standard deviation is four kg. If the analyst sets an upper control limit of 24 and a lower control limit of 20 around the target value of twenty-two kg, what is the probability of concluding that this process is out of control when it is actually in control? (n=4) UCL = 24 = 22 + z 4 4 z =1 These control limits are one standard error away from the centerline, and thus include percent of the area under the normal distribution. There is therefore a percent chance that, when the process is operating in control, a sample will indicate otherwise. 202 / 469 TRUE/FALSE An improvement in quality must necessarily increase costs. 2. TQM is important because quality influences all of the ten decisions made by operations managers. 3. Some degree of variability is present in almost all processes. 4. The purpose of process control is to detect when natural causes of variation are present. 5. Mistakes stemming from workers inadequate training represent an assignable cause of variation. 6. Averages of small samples, not individual measurements, are generally used in statistical process control. 7. X-bar charts are used when we are sampling attributes. 8. To measure the voltage of batteries, one would sample by attributes. 9. A p-chart is appropriate to plot the number of typographic errors per page of text. 10. A c-chart is appropriate to plot the number of flaws in a bolt of fabric. 11. A process that is in statistical control will always yield products that meet their design specifications. 12. The higher the process capability ratio, the greater the likelihood that process will be within design specifications. 14 1)F 2)T 3)T 4)F 5)T 6)T 7)F 8)F 9)F 10)T 11)F 12)T 203 / 469