Quantitative Analysis for Management

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(1)S e v e n t h. Quantitative Analysis for Management Barry Render Ralph M. Stair, Jr. MODULES M1 M2 M3 M4 M5 M6. Statistical Quality Control Dynamic Programming Decision Theory and the Normal Distribution Material Requirements and Just-in-Time Inventory Mathematical Tools: Determinants and Matrices The Binomial Distribution. E d i t i o n.

(2) M O D U L E. 1. Statistical Quality Control. LEARNING OBJECTIVES After completing this module, students will be able to: 1. Define the quality of a product or service. 2. Develop four types of control charts: x , R, p, and c. 3. Understand the basic theoretical underpinnings of statistical quality control, including the central limit theorem.. 4. Know if a process is in control or not.. MODULE OUTLINE M1.1 M1.2 M1.3 M1.4 M1.5. Introduction Defining Quality and TQM Statistical Process Control Control Charts for Variables Control Charts for Attributes Summary • Glossary • Key Equations • Solved Problems • Self-Test • Discussion Questions and Problems • Case Study: Bayfield Mud Company • Case Study: Morristown Daily Tribune • Bibliography. Appendix M1.1: Using QM for Windows for SPC. M1-1.

(3) M1-2. Module 1. STATISTICAL QUALITY CONTROL. M1.1 INTRODUCTION ●. Statistical process control uses statistical and probability tools to help control processes and produce consistent goods and services.. The quality of a product or service is the degree to which the product or service meets specifications.. Total Quality Management encompasses the whole organization.. For almost every product or service, there is more than one organization trying to make a sale. Price may be a major issue in whether a sale is made or lost, but another factor is quality. In fact, quality is often the major issue; and poor quality can be very expensive for both the producing firm and the customer. Consequently, firms employ quality management tactics. Quality management, or as it is more commonly called, quality control (QC), is critical throughout the organization. One of the manager’s major roles is to ensure that his or her firm can deliver a quality product at the right place, at the right time, and at the right price. Quality is not just of concern for manufactured products either; it is also important in services, from banking to hospital care to education. We begin this module with an attempt to define just what quality really is. Then we deal with the most important statistical methodology for quality management: statistical process control (SPC). SPC is the application of the statistical tools we discussed in Chapter 2 to the control of processes that result in products or services.. M1.2 DEFINING QUALITY AND TQM ● To some people, a high-quality product is one that is stronger, will last longer, is built heavier, and is, in general, more durable than other products. In some cases this is a good definition of a quality product, but not always. A good circuit breaker, for example, is not one that lasts longer during periods of high current or voltage. So the quality of a product or service is the degree to which the product or service meets specifications. Increasingly, definitions of quality include an added emphasis on meeting the customer’s needs. As you can see in Table M1.1, the first and second ones are similar to our definition. Total quality management (TQM) refers to a quality emphasis that encompasses the entire organization, from supplier to customer. TQM emphasizes a commitment by man-. T A B L E M 1 . 1 Several Definitions of Quality “Quality is the degree to which a specific product conforms to a design or specification.” H. L. Gilmore, “Product Conformance Cost,” Quality Progress (June 1974): 16. “Quality is the totality of features and characteristics of a product or service that bears on its ability to satisfy stated or implied needs.” Ross Johnson and William O. Winchell, Production and Quality, Milwaukee, WI: American Society of Quality Control, 1989, p. 2. “Quality is fitness for use.” J. M. Juran, ed., Quality Control Handbook, 3rd ed., New York: McGraw-Hill, 1974, p. 2. “Quality is defined by the customer; customers want products and services that, throughout their lives, meet customers’ needs and expectations at a cost that represents value.” Ford’s definition as presented in William W. Scherkenbach, Deming’s Road to Continual Improvement, Knoxville, TN: SPC Press, 1991, p. 161. “Even though quality cannot be defined, you know what it is.” R. M. Pirsig, Zen and the Art of Motorcycle Maintenance, New York: Bantam Books, 1974, p. 213..

(4) M1.3 Statistical Process Control. HISTORY. M1-3. How Quality Control Has Evolved. In the early nineteenth century an individual skilled artisan started and finished a whole product. With the Industrial Revolution and the factory system, semiskilled workers, each making a small portion of the final product, became common. With this, responsibility for the quality of the final product tended to shift to supervisors, and pride of workmanship declined. As organizations became larger in the twentieth century, inspection became more technical and organized. Inspectors were often grouped together; their job was to make sure that bad lots were not shipped to customers. Starting in the 1920s, major statistical QC tools were developed. W. Shewhart introduced control charts in 1924, and in 1930 H. F. Dodge and H. G. Romig designed acceptance sampling tables. Also at that time the important role of quality control in all areas of the company’s performance became recognized.. During and after World War II, the importance of quality grew, often with the encouragement of the U.S. government. Companies recognized that more than just inspection was needed to make a quality product. Quality needed to be built into the production process. After World War II, an American, W. Edwards Deming, went to Japan to teach statistical quality control concepts to the devastated Japanese manufacturing sector. A second pioneer, J. M. Juran, followed Deming to Japan, stressing top management support and involvement in the quality battle. In 1961 A. V. Feigenbaum wrote his classic book Total Quality Control, which delivered a fundamental message: Make it right the first time! In 1979 Philip Crosby published Quality Is Free, stressing the need for management and employee commitment to the battle against poor quality. In 1988, the U.S. government presented its first awards for quality achievement. These are known as the Malcolm Baldrige National Quality Awards.. agement to have a companywide drive toward excellence in all aspects of the products and services that are important to the customer. Meeting the customer’s expectations requires an emphasis on TQM if the firm is to compete as a leader in world markets.. M1.3 STATISTICAL PROCESS CONTROL ● Statistical process control (SPC) is concerned with establishing standards, monitoring standards, making measurements, and taking corrective action as a product or service is being produced. Samples of process outputs are examined; if they are within acceptable limits, the process is permitted to continue. If they fall outside certain specific ranges, the process is stopped and, typically, the assignable cause is located and removed. Control charts are graphs that show upper and lower limits for the process we want to control. A control chart is a graphic presentation of data over time. Control charts are constructed in such a way that new data can quickly be compared to past performance. Upper and lower limits in a control chart can be in units of temperature, pressure, weight, length, and so on. We take samples of the process output and plot the average of these samples on a chart that has the limits on it. Figure M1.1 graphically reveals the useful information that can be portrayed in control charts. When the average of the samples falls within the upper and lower control limits and no discernible pattern is present, the process is said to be in control; otherwise, the process is out of control or out of adjustment.. Variability in the Process All processes are subject to a certain degree of variability. Walter Shewhart of Bell Laboratories, while studying process data in the 1920s, made the distinction between the common and special causes of variation. The key is keeping variations under control. So we now look at how to build control charts that help managers and workers develop a process that is capable of producing within established limits.. SPC helps set standards. It can also monitor, measure, and correct quality problems.. A control chart is a graphic way of presenting data over time..

(5) M1-4. Module 1. FIGURE M1.1 Patterns to Look for on Control Charts (Source: Bertrand L. Hansen, Quality Control: Theory and Applications, © 1963, renewed 1991, p. 65. Reprinted by permission of Prentice Hall, Upper Saddle River, NJ.). STATISTICAL QUALITY CONTROL. Upper control limit. Target. Lower control limit Normal behavior.. One plot out above. Investigate for cause.. One plot out below. Investigate for cause.. Two plots near upper control. Investigate for cause.. Two plots near lower control. Investigate for cause.. Run of 5 above central line. Investigate for cause.. Upper control limit. Target. Lower control limit. Upper control limit. Target. Lower control limit Run of 5 below central line. Trends in either direction 5 Investigate for cause. plots. Investigate for cause of progressive change.. Erratic behavior. Investigate.. Building Control Charts When building control charts, averages of small samples (often of five items or parts) are used, as opposed to data on individual parts. Individual pieces tend to be too erratic to make trends quickly visible. The purpose of control charts is to help distinguish between natural variations and variations due to assignable causes.. IN ACTION. Statistical Process Control Helps Du Pont and the Environment. Du Pont has found that statistical process control (SPC) is an excellent approach to solving environmental problems. With a goal of slashing manufacturing waste and hazardous waste disposals by 35%, Du Pont brought together information from its quality control systems and its material management databases. Diagrams and charts examining causes and effects revealed where major problems occurred. Then the company began reducing waste materials through improved SPC standards for production. Tying together shop-floor informationbased monitoring systems with air-quality standards, Du Pont identified ways to reduce emissions. Using a vendor evaluation system linked to just-in-time purchasing requirements, the company initiated controls over incoming hazardous materials.. Du Pont now saves more than 15 million pounds of plastics annually by recycling them into products rather than dumping them into landfills. Through electronic purchasing the firm has reduced wastepaper to a trickle and, by using new packaging designs, has cut in-process material wastes by nearly 40%. By integrating SPC with environmental compliance activities, Du Pont has made major quality improvements that far exceed regulatory guidelines, and at the same time the company has realized huge cost savings. Sources: Automotive Industries (June 1996): 93; and E. E. Dwinells and J. P. Sheffer, APICS—The Performance Advantage (March 1992): 30–31..

(6) M1.4 Control Charts for Variables. Natural Variations Natural variations affect almost every production process and are to be expected. Natural variations are the many sources of variation within a process that is in statistical control. They behave like a constant system of chance causes. Although individual measured values are all different, as a group they form a pattern that can be described as a distribution. When these distributions are normal, they are characterized by two parameters. These parameters are. M1-5. Natural variations are sources of variation in a process that is statistically in control.. 1. Mean, (the measure of central tendency, in this case, the average value) 2. Standard deviation, (variation, the amount by which the smaller values differ from the larger ones) As long as the distribution (output precision) remains within specified limits, the process is said to be “in control,” and the modest variations are tolerated. Assignable Variations When a process is not in control, we must detect and eliminate special (assignable) causes of variation. Then its performance is predictable, and its ability to meet customer expectations can be assessed. The ability of a process to operate within statistical control is determined by the total variation that comes from natural causes—the minimum variation that can be achieved after all assignable causes have been eliminated. The objective of a process control system, then, is to provide a statistical signal when assignable causes of variation are present. Such a signal can quicken appropriate action to eliminate assignable causes. Assignable variation in a process can be traced to a specific reason. Factors such as machine wear, misadjusted equipment, fatigued or untrained workers, or new batches of raw material are all potential sources of assignable variations. Control charts such as those illustrated in Figure M1.1 help the manager pinpont where a problem may lie.. Assignable variations in a process can be traced to a specific problem.. M1.4 CONTROL CHARTS FOR VARIABLES ● Control charts for the mean, x, and the range, R, are used to monitor processes that are measured in continuous units. The x- (x-bar) chart tells us whether changes have occurred in the central tendency of a process. This might be due to such factors as tool wear, a gradual increase in temperature, a different method used on the second shift, or new and stronger materials. The R-chart values indicate that a gain or loss in uniformity has occurred. Such a change might be due to worn bearings, a loose tool part, an erratic flow of lubricants to a machine, or to sloppiness on the part of a machine operator. The two types of charts go hand in hand when monitoring variables.. x-charts measure central tendency of a process.. R-charts measure the range between the biggest (or heaviest) and smallest (or lightest) items in a random sample.. The Central Limit Theorem The statistical foundation for x-charts is the central limit theorem. In general terms, this theorem states that regardless of the distribution of the population of all parts or services, the distribution of 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 large. Fortunately, even if n is fairly small (say 4 or 5), the distributions of the averages will still roughly follow a normal curve. The theorem also states that (1) the mean of the distribution of the x’s (called x) will equal the mean of the overall population (called ); and (2) the standard deviation of the sampling distribution, sx, will be the population deviation, x, divided by the square root of the sample size, n. In other words, xm. and. sx . sx n. The central limit theorem says that the distribution of sample means will follow a normal distribution as the sample size grows large..

(7) M1-6. Module 1. FIGURE M1.2 Population and Sampling Distributions. STATISTICAL QUALITY CONTROL. Some population distributions. Normal. Beta. (mean) x S.D.. Uniform. (mean) x S.D.. (mean) x S.D.. Sampling distribution of sample means (always normal). 99.7% of all x fall within ±3x 95.5% of all x fall within ±2x. 3x. 2x. 1x. X (mean). Standard error x . 1x. 2x. 3x. x n . Figure M1.2 shows three possible population distributions, each with its own mean, , and standard deviation, x. If a series of random samples (x1, x2, x3, x4, and so on) each of size n is drawn from any one of these, the resulting distribution of xi’s will appear as in the bottom graph of that figure. Because this is a normal distribution (as discussed in Chapter 2), we can state that 1. 99.7% of the time, the sample averages will fall within 3sx if the process has only random variations. 2. 95.5% of the time, the sample averages will fall within 2sx if the process has only random variations. If a point on the control chart falls outside the 3sx control limits, we are 99.7% sure that the process has changed. This is the theory behind control charts.. Setting x -Chart Limits If we know through historical data the standard deviation of the process population, sx, we can set upper and lower control limits by these formulas: upper control limit (ULC) x zsx. (M1-1). lower control limit (LCL) x zsx. (M1-2).

(8) M1.4 Control Charts for Variables. M1-7. where x mean of the sample means z number of normal standard deviations (2 for 95.5% confidence, 3 for 99.7%) s sx standard deviation of the sample means x n Box-Filling Example Let us say that a large production lot of boxes of cornflakes is sampled every hour. To set control limits that include 99.7% of the sample means, 36 boxes are randomly selected and weighed. The standard deviation of the overall population of boxes is estimated, through analysis of old records, to be 2 ounces. The average mean of all samples taken is 16 ounces. We therefore have x 16 ounces, x 2 ounces, n 36, and z 3. The control limits are 2 36 16 1 17 ounces 2 16 3 16 1 15 ounces 36. UCL x x zsx 16 3 LCL x x zsx. MODELING IN THE REAL WORLD Defining the Problem. Developing a Model. Statistical Process Control at AVX-Kyocera. AVX-Kyocera, a Japanese-owned maker of electronic chip components located in Raleigh, North Carolina, needed to improve the quality of its products and services to achieve total customer satisfaction.. Statistical process control models such as x- and R-charts were chosen as appropriate tools.. Acquiring Input Data. Employees are empowered to collect their own data. For example, a casting machine operator measures the thickness of periodic samples that he takes from his process.. Developing a Solution. Employees plot data observations to generate SPC charts that track trends, comparing results with process limits and final customer specifications.. Testing the Solution. Samples at each machine are evaluated to ensure that the processes are indeed capable of achieving the desired results. Quality control inspectors are transferred to manufacturing duties as all plant personnel become trained in statistical methodology.. Analyzing the Results. Implementing the Results. Results of SPC are analyzed by individual operators to see if trends are present in their processes. Quality trend boards are posted in the building to display not only SPC charts, but also procedures, process document change approvals, and the names of all certified operators. Work teams are in charge of analysis of clusters of machines. The firm has implemented a policy of zero defectives at a very low tolerance for variable data and nearly zero defects for parts per million for attribute data. Source: Basile A. Denisson. “War with Defects and Peace with Quality,” Quality Progress (September 1993): 97–101..

(9) M1-8. Module 1. STATISTICAL QUALITY CONTROL. If the process standard deviation is not available or is difficult to compute, which is usually the case, these equations become impractical. In practice, the calculation of control limits is based on the average range rather than on standard deviations. We may use the equations Control chart limits can be found using the range rather than the standard deviation.. UCL x x A2R. (M1-3). LCL x x A2R. (M1-4). where R average of the samples A2 value found in Table M1.2 (which assumes that Z 3) x mean of the sample means Here is an example. Super Cola bottles soft drinks labeled “net weight 16 ounces.” An overall process average of 16.01 ounces has been found by taking several batches of samples, where each sample contained five bottles. The average range of the process is 0.25 ounce. We want to determine the upper and lower control limits for averages for this process. Looking in Table M1.2 for a sample size of 5 in the mean factor A2 column, we find the number 0.577. Thus the upper and lower control chart limits are UCLx x A2R 16.01 (0.577)(0.25) 16.01 0.144 16.154 LCLx x A2R 16.01 0.144 15.866 The upper control limit is 16.154, and the lower control limit is 15.866.. Setting Range Chart Limits Dispersion or variability is also important. The central tendency can be under control, but ranges can be out of control.. We just determined the upper and lower control limits for the process average. In addition to being concerned with the process average, managers are interested in the dispersion or variability. Even though the process average is under control, the variability of the process may not be. For example, something may have worked itself loose in a piece of equipment. As a result, the average of the samples may remain the same, but the variation within the samples could be entirely too large. For this reason it is very common to find a control chart for ranges in order to monitor the process variability. The theory behind the control charts for ranges is the same for the process average. Limits are established that contain 3 standard deviations of the distribution for the average range R. With a few simplifying assumptions, we can set the upper and lower control limits for ranges: UCLR D4R LCLR D3R where UCLR upper control chart limit for the range LCLR lower control chart limit for the range D4 and D3 values from Table M1.2. (M1-5) (M1-6).

(10) M1.5 Control Charts for Attributes. M1-9. T A B L E M 1 . 2 Factors for Computing Control Chart Limits SAMPLE SIZE, n. MEAN FACTOR, A2. UPPER RANGE, D4. LOWER RANGE, D3. 2. 1.880. 3.268. 0. 3. 1.023. 2.574. 0. 4. 0.729. 2.282. 0. 5. 0.577. 2.114. 0. 6. 0.483. 2.004. 0. 7. 0.419. 1.924. 0.076. 8. 0.373. 1.864. 0.136. 9. 0.337. 1.816. 0.184. 10. 0.308. 1.777. 0.223. 12. 0.266. 1.716. 0.284. 14. 0.235. 1.671. 0.329. 16. 0.212. 1.636. 0.364. 18. 0.194. 1.608. 0.392. 20. 0.180. 1.586. 0.414. 25. 0.153. 1.541. 0.459. Source: Reprinted by permission of the American Society for Testing and Materials, copyright. Taken from Special Technical Publication 15-C, “Quality Control of Materials,” pp. 63 and 72, 1951.. Range Example As an example, 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 chart limits. Looking in Table M1.2 for a sample size of 5, we find that D4 2.114 and D3 0. The range control chart limits are UCLR D4R (2.114)(53 pounds) 112.042 pounds LCLR D3R (0)(53 pounds) 0. M1.5 CONTROL CHARTS FOR ATTRIBUTES ● Control charts for x and R do not apply when we are sampling attributes, which are typically classified as defective or nondefective. Measuring defectives involves counting them (for example, number of bad lightbulbs in a given lot, or number of letters or data entry. Sampling attributes differ from sampling variables..

(11) M1-10. Module 1. STATISTICAL QUALITY CONTROL. Five Steps to Follow in Using x and R-Charts 1. Collect 20 to 25 samples of n 4 or n 5 each from a stable process and compute the mean and range of each. 2. Compute the overall means (x and R), set appropriate control limits, usually at the 99.7% level, and calculate the preliminary upper and lower control limits. If the process is not currently stable, use the desired mean, , instead of x to calculate limits. 3. Graph the sample means and ranges on their respective control charts and determine whether they fall outside the acceptable limits. 4. Investigate points or patterns that indicate the process is out of control. Try to assign cases for the variation and then resume the process. 5. Collect additional samples and, if necessary, revalidate the control limits using the new data.. records typed with errors); whereas variables are usually measured for length or weight. There are two kinds of attribute control charts: (1) those that measure the percent defective in a sample, called p-charts, and (2) those that count the number of defects, called c-charts.. p-Charts p-chart limits are based on the binomial distribution and are easy to compute.. p-charts are the principal means of controlling attributes. Although attributes that are either good or bad follow the binomial distribution, the normal distribution can be used to calculate p-chart limits when sample sizes are large. The procedure resembles the x-chart approach, which was also based on the central limit theorem. The formulas for p-chart upper and lower control limits follow: UCLp p zsp. (M1-7). LCLp p zsp. (M1-8). where p mean fraction defective in the sample z number of standard deviates (z = 2 for 95.5% limits; z = 3 for 99.7% limits) sp standard deviation of the sampling distribution. p is estimated by the formula sp . . p(1 p) n. (M1-9). where n is the size of each sample. ARCO p-Chart Example Using a popular database software package, data-entry clerks at ARCO key in thousands of insurance records each day. Samples of the work of 20 clerks are shown in the following table. One hundred records entered by each clerk were carefully examined to make sure that they contained no errors; the fraction defective in each sample was then computed..

(12) M1.5 Control Charts for Attributes. SAMPLE NUMBER. NUMBER OF ERRORS. FRACTION DEFECTIVE. SAMPLE NUMBER. NUMBER OF ERRORS. 1. 6. 0.06. 11. 6. 0.06. 2. 5. 0.05. 12. 1. 0.01. 3. 0. 0.00. 13. 8. 0.08. 4. 1. 0.01. 14. 7. 0.07. 5. 4. 0.04. 15. 5. 0.05. 6. 2. 0.02. 16. 4. 0.04. 7. 5. 0.05. 17. 11. 0.11. 8. 3. 0.03. 18. 3. 0.03. 9. 3. 0.03. 19. 0. 0.00. 10. 2. 0.02. 20. 04. 0.04. M1-11. FRACTION DEFECTIVE. 80. We want to set control limits that include 99.7% of the random variation in the entry process when it is in control. Thus, z 3. p. (0.04)(1100 0.04) 0.02 a. sp . total number of errors 80 0.04 total number of records examined (100)(20). (Note: 100 is the size of each sample n) LCLp p zsp 0.04 3(0.02) 0. a. UCLp p zsp 0.04 3(0.02) 0.10. (since we cannot have a negative percent defective) When we plot the control limits and the sample fraction defectives, we find that only one data-entry clerk (number 17) is out of control. The firm may wish to examine that person’s work a bit more closely to see whether a serious problem exists (see Figure M1.3). Using Excel QM for SPC Excel and other spreadsheets are extensively used in industry to maintain control charts. Excel QM’s Quality Module has the ability to develop xcharts, p-charts, and c-charts. Programs M1.1a and M1.1b illustrate Excel QM’s spreadsheet approach to computing the p-chart control limits for the ARCO example. Program M1.1a shows both the data input and formulas. Program M1.1b provides output. Excel also contains a built-in graphing ability with Chart Wizard.. c-Charts In the ARCO example above, we counted the number of defective database records entered. A defective record was one that was not exactly correct. A bad record may contain more than one defect, however. We use c-charts to control the number of defects per unit of output (or per insurance record in the case above).. c-charts count the number of defects, whereas p-charts track the percentage defective..

(13) M1-12. Module 1. STATISTICAL QUALITY CONTROL. Fraction defective. FIGURE M1.3 p-Chart for Data Entry for ARCO 0.12 0.11 0.10 0.09 0.08 0.07. UCLp 0.10. 0.06 0.05 0.04 0.03 0.02. p 0.04. 0.01. LCLp 0.00. 0.00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Sample number. Control charts for defects are helpful for monitoring processes in which a large number of potential errors can occur but the actual number that do occur is relatively small. Defects may be mistyped words in a newspaper, blemishes on a table, or missing pickles on a fast-food hamburger. The Poisson probability distribution, which has a variance equal to its mean, is the basis for c-charts. Since c is the mean number of defects per unit, the standard deviation is equal to c. To compute 99.7% control limits for c, we use the formula c 3c PROGRAM M1.1A Excel QM’s p-Chart Program Applied to the ARCO Data, Showing Input Data and Formulas. (M1-10).

(14) Summary. M1-13. PROGRAM M1.1B Output from Excel QM’s p-Chart Analysis of the ARCO Data. Here is an example. Red Top Cab Company c-Chart Example Red Top Cab Company receives several complaints per day about the behavior of its drivers. Over a nine-day period (where days are the units of measure), the owner received the following numbers of calls from irate passengers: 3, 0, 8, 9, 6, 7, 4, 9, 8, for a total of 54 complaints. To compute 99.7% control limits, we take c. 54 6 complaints per day 9. Thus, LCLc c 3c 6 36 6 3(2.45) 0. a. UCLc c 3c 6 36 6 3(2.45) 13.35. (because we cannot have a negative control limit) After the owner plotted a control chart summarizing these data and posted it prominently in the drivers’ locker room, the number of calls received dropped to an average of three per day. Can you explain why this may have occurred?. Summary To the manager of a firm producing goods or services, quality is the degree to which the product meets specifications. Quality control has become one of the most important precepts of business. The expression “quality cannot be inspected into a product” is a central theme of organizations today. More and more world-class companies are following the ideas of total quality management (TQM), which emphasizes the entire organization, from supplier to customer..

(15) M1-14. Module 1. STATISTICAL QUALITY CONTROL. Statistical aspects of quality control date to the 1920s but are of special interest in our global marketplaces of this new century. Statistical process control (SPC) tools described in this chapter include the x- and R-charts for variable sampling and the p- and c-charts for attribute sampling.. Glossary Quality. The degree to which a product or service meets the specifications set for it. Total Quality Management (TQM). An emphasis on quality that encompasses the entire organization. Control Chart. A graphic presentation of process data over time. Natural Variations. Variabilities that affect almost every production process to some degree and are to be expected; also known as common causes. Assignable Variation. Variation in the production process that can be traced to specific causes. x-Chart. A quality control chart for variables that indicates when changes occur in the central tendency of a production process. R-Chart. A process control chart that tracks the “range” within a sample; indicates that a gain or loss of uniformity has occurred in a production process. Central Limit Theorem. The theoretical foundation for x-charts. It states that regardless of the distribution of the population of all parts or services, the distribution of x’s will tend to follow a normal curve as the sample size grows. p-Chart. A quality control chart that is used to control attributes. c-Chart. A quality control chart that is used to control the number of defects per unit of output.. Key Equations (M1-1) Upper control limit (UCL) x Zsx The upper limit for an x-chart using standard deviations. (M1-2) Lower control limit (LCL) x Zsx The lower control limit for an x-chart using standard deviations. (M1-3) UCLx x A2R The upper control limit for an x-chart using tabled values and ranges. (M1-4) LCLx x A2R The lower control limit for an x-chart using tabled values and ranges. (M1-5) UCLR D4R Upper control limit for a range chart. (M1-6) LCLR D3R Lower control limit for a range chart. (M1-7) UCLp p zsp Upper control unit for a p-chart. (M1-8) LCLp p zsp Lower control limit for a p-chart. p(1 p) (M1-9) sp n. . The standard deviation of a binomial distribution. (M1-10) c 3c The upper and lower limits for a c-chart..

(16) Solved Problems. Solved Problems Solved Problem M1-1 The manufacturer of precision parts for drill presses produces round shafts for use in the construction of drill presses. The average diameter of a shaft is 0.56 inch. The inspection samples contain six shafts each. The average range of these samples is 0.006 inch. Determine the upper and lower control chart limits. Solution The mean factor A2 from Table M1.2, where the sample size is 6, is seen to be 0.483. With this factor, you can obtain the upper and lower control limits: UCLx 0.56 (0.483)(0.006). LCLx 0.56 0.0029. 0.56 0.0029 0.5629. 0.5571. Solved Problem M1-2 Nocaf Drinks, Inc., a producer of decaffeinated coffee, bottles Nocaf. Each bottle should have a net weight of 4 ounces. The machine that fills the bottles with coffee is new, and the operations manager wants to make sure that it is properly adjusted. The operations manager takes a sample of n 8 bottles and records the average and range in ounces for each sample. The data for several samples are given in the following table. Note that every sample consists of 8 bottles.. SAMPLE. SAMPLE RANGE. SAMPLE AVERAGE. SAMPLE. SAMPLE RANGE. SAMPLE AVERAGE. A. 0.41. 4.00. E. 0.56. 4.17. B. 0.55. 4.16. F. 0.62. 3.93. C. 0.44. 3.99. G. 0.54. 3.98. D. 0.48. 4.00. H. 0.44. 4.01. Is the machine properly adjusted and in control? Solution We first find that x 4.03 and R 0.51. Then, using Table M1.2, we find UCLx x A2R 4.03 (0.373)(0.51) 4.22 LCLx x A2R 4.03 (0.373)(0.51) 3.84 UCLR D4R (1.864)(0.51) 0.95 LCLR D3R (0.136)(0.51) 0.07 It appears that the process average and range are both in control. Solved Problem M1-3 Crabill Electronics, Inc., makes resistors, and among the last 100 resistors inspected, the percent defective has been 0.05. Determine the upper and lower limits for this process for 99.7% confidence.. M1-15.

(17) M1-16. Module 1. STATISTICAL QUALITY CONTROL. Solution. . UCLp p 3. . p(1 p) (0.05)(1 0.05) 0.05 3 n 100. 0.05 3(0.0218) 0.1154. . LCLp p 3. p(1 p) 0.05 3(0.0218) n. 0.05 0.0654 0 (since percent defective cannot be negative).

(18) Self-Test. M1-17. SELF-TEST • • •. Before taking the self-test, refer back to the learning objectives at the beginning of the module, the notes in the margins, and the glossary at the end of the module. Use the key at the back of the book to correct your answers. Restudy pages that correspond to any questions that you answered incorrectly or material you feel uncertain about.. 1. Quality is defined as ________________ . 2. A control chart is a. a means of monitoring output. b. a graphic presentation of data over time. c. a chart with upper and lower control limits. d. all of the above. e. none of the above. 3. The type of chart used to control the number of defects per unit of output is the a. x-bar chart. b. R-chart. c. p-chart. d. all of the above. e. none of the above. 4. Control charts for attributes are a. p-charts. b. m-charts. c. R-charts. d. x-charts. e. none of the above. 5. c-Charts are based on the a. Poisson distribution. b. normal distribution. c. Erlang distribution. d. hyper Erlang distribution. e. binomial distribution. f. none of the above. 6. If a sample of parts are measured and the mean of the sample measurement is outside the tolerance limits, a. the process is out of control and the cause can be established. b. the process is in control but is not capable of producing within the established control limits.. 7.. 8.. 9. 10.. 11.. c. the process is within the established control limits with only natural causes of variation. d. all of the above are true. e. none of the above are true. If a sample of parts are measured and the mean of the sample measurement is in the middle of the tolerance limits but some parts measure too low and other parts measure too high, a. the process is out of control and the cause can be established. b. the process is in control but is not capable of producing within the established control limits. c. the process is within the established control limits with only natural causes of variation. d. all of the above are true. e. none of the above are true. A goal of 6 means expecting a. 2,700 errors per million parts. b. 95.45% accuracy. c. 99.73% of errors are caught. d. 3.4 errors per million parts. e. none of the above. A range chart monitors ________________ . If a 95.5% level of confidence is desired, the x-chart limits will be set plus or minus ________________ . The two techniques discussed to find and resolve assignable variations in process control are the ______________ and the ______________ ..

(19) M1-18. Module 1. STATISTICAL QUALITY CONTROL. Discussion Questions and Problems Discussion Questions M1-1. Why is the central limit theorem so important in statistical quality control?. M1-2. Why are x- and R-charts usually used hand in hand?. M1-3. Explain the differences among the four types of control charts.. M1-4. What might cause a process to be out of control?. M1-5. Explain why a process can be out of control even though all the samples fall within the upper and lower control limits.. Problems* • M1-6 Shader Storage Technologies produces refrigeration units for food producers and retail food establishments. The overall average temperature that these units maintain is 46 Fahrenheit. The average range is 2 Fahrenheit. Samples of 6 are taken to monitor the process. Determine the upper and lower control chart limits for averages and ranges for these refrigeration units.. • M1-7. • M1-8. •• M1-9. When set at the standard position, Autopitch can throw hard balls toward a batter at an average speed of 60 mph. Autopitch devices are made for both major- and minorleague teams to help them improve their batting averages. Autopitch executives take samples of 10 Autopitch devices at a time to monitor these devices and to maintain the highest quality. The average range is 3 mph. Using control-chart techniques, determine control-chart limits for averages and ranges for Autopitch. Zipper Products, Inc., produces granola cereal, granola bars, and other natural food products. Its natural granola cereal is sampled to ensure proper weight. Each sample contains eight boxes of cereal. The overall average for the samples is 17 ounces. The range is only 0.5 ounce. Determine the upper and lower control-chart limits for averages for the boxes of cereal. Small boxes of NutraFlakes cereal are labeled “net weight 10 ounces.” Each hour, random samples of size n 4 boxes are weighed to check process control. Five hours of observations yielded the following:. WEIGHT Time. * Note:. Box 1. Box 2. Box 3. Box 4. 9 A.M.. 9.8. 10.4. 9.9. 10.3. 10 A.M.. 10.1. 10.2. 9.9. 9.8. 11 A.M.. 9.9. 10.5. 10.3. 10.1. Noon. 9.7. 9.8. 10.3. 10.2. 1 A.M.. 9.7. 10.1. 9.9. 9.9. means the problem may be solved with QM for Windows;. be solved with Excel QM; and and/or Excel QM.. means the problem may. means the problem may be solved with QM for Windows.

(20) Discussion Questions and Problems. Using these data, construct limits for x- and R-charts. Is the process in control? What other steps should the QC department follow at this point?. •• M1-10 Sampling four pieces of precision-cut wire (to be used in computer assembly) every hour for the past 24 hours has produced the following results:. HOUR. x. R. HOUR. x. R. 1. 3.25. 0.71. 13. 3.11. 0.85. 2. 3.10. 1.18. 14. 2.83. 1.31. 3. 3.22. 1.43. 15. 3.12. 1.06. 4. 3.39. 1.26. 16. 2.84. 0.50. 5. 3.07. 1.17. 17. 2.86. 1.43. 6. 2.86. 0.32. 18. 2.74. 1.29. 7. 3.05. 0.53. 19. 3.41. 1.61. 8. 2.65. 1.13. 20. 2.89. 1.09. 9. 3.02. 0.71. 21. 2.65. 1.08. 10. 2.85. 1.33. 22. 3.28. 0.46. 11. 2.83. 1.17. 23. 2.94. 1.58. 12. 2.97. 0.40. 24. 2.64. 0.97. Develop appropriate control limits and determine whether there is any cause for concern in the cutting process.. •• M1-11 Due to the poor quality of various semiconductor products used in their manufacturing process, Microlaboratories has decided to develop a quality control program. Because the semiconductor parts they get from suppliers are either good or defective, Milton Fisher has decided to develop control charts for attributes. The total number of semiconductors in every sample is 200. Furthermore, Milton would like to determine the upper control chart limit and the lower control chart limit for various values of the fraction defective (p) in the sample taken. To allow more flexibility, he has decided to develop a table that lists value for p, UCL, and LCL. The values for p should range from 0.01 to 0.1, incrementing by 0.01 each time. What are the UCLs and the LCLs for 99.7% confidence?. •• M1-12 For the past two months, Suzan Shader has been concerned about machine number 5 at the West Factory. To make sure that the machine is operating correctly, samples are taken, and the average and range for each sample is computed. Each sample consists of 12 items produced from the machine. Recently, 12 samples were taken, and for each, the sample range and average were computed. The sample range and sample average were 1.1 and 46 for the first sample, 1.31 and 45 for the second sample, 0.91 and 46 for the third sample, and 1.1 and 47 for the fourth sample. After the fourth sample, the sample averages increased. For the fifth sample, the range was 1.21 and the average was 48; for sample number 6 it was 0.82 and 47; for sample number 7, it was 0.86 and 50; and for the eighth sample, it was 1.11 and 49. After the eighth sample, the sample average continued to increase, never getting below 50. For sample number 9, the range and average were 1.12 and 51; for sample number 10, they were. M1-19.

(21) M1-20. Module 1. STATISTICAL QUALITY CONTROL. 0.99 and 52; for sample number 11, they were 0.86 and 50; and for sample number 12, they were 1.2 and 52. Although Suzan’s boss wasn’t overly concerned about the process, Suzan was. During installation, the supplier set a value of 47 for the process average with an average range of 1.0. It was Suzan’s feeling that something was definitely wrong with machine number 5. Do you agree?. •• M1-13 Kitty Products caters to the growing market for cat supplies, with a full line of products, ranging from litter to toys to flea powder. One of its newer products, a tube of fluid that prevents hair balls in long-haired cats, is produced by an automated machine that is set to fill each tube with 63.5 grams of paste. To keep this filling process under control, four tubes are pulled randomly from the assembly line every 4 hours. After several days, the data shown in the following table resulted. Set control limits for this process and graph the sample data for both the xand R-charts.. Sample No.. x. R. Sample No.. x. R. Sample No.. x. R. 1. 63.5. 2.0. 10. 63.5. 1.3. 18. 63.6. 1.8. 2. 63.6. 1.0. 11. 63.3. 1.8. 19. 63.8. 1.3. 3. 63.7. 1.7. 12. 63.2. 1.0. 20. 63.5. 1.6. 4. 63.9. 0.9. 13. 63.6. 1.8. 21. 63.9. 1.0. 5. 63.4. 1.2. 14. 63.3. 1.5. 22. 63.2. 1.8. 6. 63.0. 1.6. 15. 63.4. 1.7. 23. 63.3. 1.7. 7. 63.2. 1.8. 16. 63.4. 1.4. 24. 64.0. 2.0. 8. 63.3. 1.3. 17. 63.5. 1.1. 25. 63.4. 1.5. 9. 63.7. 1.6. • M1-14 The smallest defect in a computer chip will render the entire chip worthless. Therefore, tight quality control measures must be established to monitor the chips. In the past, the percentage defective for these chips for a California-based company has been 1.1%. The sample size is 1,000. Determine the upper and lower control-chart limits for these computer chips. Use z 3.. •• M1-15 Barbara Schwartz’s Office Supply Company manufactures paper clips and other office products. Although inexpensive, paper clips have provided Barbara with a high margin of profitability. The percentage defective for paper clips produced by Office Supply Company has been averaging 2.5%. Samples of 200 paper clips are taken. Establish the upper and lower control-chart limits for this process at 99.7% confidence.. •• M1-16 Daily samples of 100 power drills are removed from Drill Master’s assembly line and inspected for defects. Over the past 21 days, the following information has been gathered. Develop a 3 standard deviation (99.7% confidence) p-chart and graph the samples. Is the process in control?.

(22) Case Study. DAY. NUMBER OF DEFECTIVE DRILLS. DAY. NUMBER OF DEFECTIVE DRILLS. 1. 6. 12. 5. 2. 5. 13. 4. 3. 6. 14. 3. 4. 4. 15. 4. 5. 3. 16. 5. 6. 4. 17. 6. 7. 5. 18. 5. 8. 3. 19. 4. 9. 6. 20. 3. 10. 3. 21. 7. 11. 7. M1-21. •M1-17 A random sample of 100 Modern Art dining room tables that came off the firm’s assembly line is examined. Careful inspection reveals a total of 2,000 blemishes. What are the 99.7% upper and lower control limits for the number of blemishes? If one table had 42 blemishes, should any special action be taken?. Case Study Bayfield Mud Company In November 1998, John Wells, a customer service representative of Bayfield Mud Company, was summoned to the Houston, Texas, warehouse of Wet-Land Drilling, Inc., to inspect three boxcars of mud treating agents that Bayfield Mud Company had shipped to the Houston firm. (Bayfield’s corporate offices and its largest plant are located in Orange, Texas, which is just west of the Louisiana—Texas border.) Wet-Land Drilling had filed a complaint that the 50-pound bags of treating agents that it had just received from Bayfield were shortweighted by approximately 5%. The light-weight bags were initially detected by one of Wet-Land’s receiving clerks, who noticed that the railroad scale tickets indicated that the net weights were significantly less on all three of the boxcars than those of identical shipments received on October 25, 1998. Bayfield’s traffic department was called to determine whether lighter-weight dunnage or pallets were used on the shipments. (This might explain the lighter net weights.) Bayfield indicated, however, that no changes had been made in the loading or palletizing procedures. Hence, Wet-Land randomly checked 50 of the bags and discovered that the average net weight was 47.51 pounds. They noted from past shipments that the bag net weights averaged exactly 50.0 pounds, with an acceptable standard deviation of. 1.2 pounds. Consequently, they concluded that the sample indicated a significant short-weight. (The reader may wish to verify this conclusion.) Bayfield was then contacted, and Wells was sent to investigate the complaint. Upon arrival, Wells verified the complaint and issued a 5% credit to Wet-Land. Wet-Land’s management, however, was not completely satisfied with only the issuance of credit for the short shipment. The charts followed by their mud engineers on the drilling platforms were based on 50-pound bags of treating agents. Lighterweight bags might result in poor chemical control during the drilling operation and might adversely affect drilling efficiency. (Mud treating agents are used to control the pH and other chemical properties of the cone during drilling operation.) This could cause severe economic consequences because of the extremely high cost of oil and natural gas well drilling operations. Consequently, special use instructions had to accompany the delivery of these shipments to the drilling platforms. Moreover, the light-weight shipments had to be isolated in Wet-Land’s warehouse, causing extra handling and poor space utilization. Hence, Wells was informed that Wet-Land Drilling might seek a new supplier of mud treating agents if, in the future, it received bags that deviated significantly from 50 pounds. The quality control department at Bayfield suspected that the light-weight bags may have resulted from “growing pains”.

(23) M1-22. Module 1. STATISTICAL QUALITY CONTROL. TIME. AVERAGE WEIGHT (POUNDS). TIME. AVERAGE WEIGHT (POUNDS). Smallest. Largest. Smallest. Largest. 6:00 A.M.. 49.6. 48.7. 50.7. 6:00 A.M.. 46.8. 41.0. 51.2. 7:00. 50.2. 8:00. 50.6. 49.1. 51.2. 7:00. 50.0. 46.2. 51.7. 49.6. 51.4. 8:00. 47.4. 44.0. 48.7. 9:00. 50.8. 50.2. 51.8. 9:00. 47.0. 44.2. 48.9. 10:00. 49.9. 49.2. 52.3. 10:00. 47.2. 46.6. 50.2. 11:00. 50.3. 48.6. 51.7. 11:00. 48.6. 47.0. 50.0. 12 noon. 48.6. 46.2. 50.4. 12 midnight. 49.8. 48.2. 50.4. 1:00 P.M.. 49.0. 46.4. 50.0. 1:00 A.M.. 49.6. 48.4. 51.7. 2:00. 49.0. 46.0. 50.6. 2:00. 50.0. 49.0. 52.2. 3:00. 49.8. 48.2. 50.8. 3:00. 50.0. 49.2. 50.0. 4:00. 50.3. 49.2. 52.7. 4:00. 47.2. 46.3. 50.5. 5:00. 51.4. 50.0. 55.3. 5:00. 47.0. 44.1. 49.7. 6:00. 51.6. 49.2. 54.7. 6:00. 48.4. 45.0. 49.0. 7:00. 51.8. 50.0. 55.6. 7:00. 48.8. 44.8. 49.7. 8:00. 51.0. 48.6. 53.2. 8:00. 49.6. 48.0. 51.8. 9:00. 50.5. 49.4. 52.4. 9:00. 50.0. 48.1. 52.7. 10:00. 49.2. 46.1. 50.7. 10:00. 51.0. 48.1. 55.2. 11:00. 49.0. 46.3. 50.8. 11:00. 50.4. 49.5. 54.1. 12 midnight. 48.4. 45.4. 50.2. 12 noon. 50.0. 48.7. 50.9. 1:00 A.M.. 47.6. 44.3. 49.7. 1:00 P.M.. 48.9. 47.6. 51.2. 2:00. 47.4. 44.1. 49.6. 2:00. 49.8. 48.4. 51.0. 3:00. 48.2. 45.2. 49.0. 3:00. 49.8. 48.8. 50.8. 4:00. 48.0. 45.5. 49.1. 4:00. 50.0. 49.1. 50.6. 5:00. 48.4. 47.1. 49.6. 5:00. 47.8. 45.2. 51.2. 6:00. 48.6. 47.4. 52.0. 6:00. 46.4. 44.0. 49.7. 7:00. 50.0. 49.2. 52.2. 7:00. 46.4. 44.4. 50.0. 8:00. 49.8. 49.0. 52.4. 8:00. 47.2. 46.6. 48.9. 9:00. 50.3. 49.4. 51.7. 9:00. 48.4. 47.2. 49.5. 10:00. 50.2. 49.6. 51.8. 10:00. 49.2. 48.1. 50.7. 11:00. 50.0. 49.0. 52.3. 11:00. 48.4. 47.0. 50.8. 12 noon. 50.0. 48.8. 52.4. 12 midnight. 47.2. 46.4. 49.2. 1:00 A.M.. 50.1. 49.4. 53.6. 1:00 A.M.. 47.4. 46.8. 49.0. 2:00. 49.7. 48.6. 51.0. 2:00. 48.8. 47.2. 51.4. 3:00. 48.4. 47.2. 51.7. 3:00. 49.6. 49.0. 50.6. 4:00. 47.2. 45.3. 50.9. 4:00. 51.0. 50.5. 51.5. 5:00. 46.8. 44.1. 49.0. 5:00. 50.5. 50.0. 51.9. RANGE. RANGE.

(24) M1-23. Case Study. at the Orange plant. Because of the earlier energy crisis, oil and natural gas exploration activity had greatly increased. This increased activity, in turn, created increased demand for products produced by related industries, including drilling muds. Consequently, Bayfield had to expand from a one-shift (6:00 A.M. to 2:00 P.M.) to a two-shift (6:00 A.M. to 10:00 P.M.) operation in mid-1996, and finally to a three-shift operation (24 hours per day) in the fall of 1998. The additional night-shift bagging crew was staffed entirely by new employees. The most experienced supervisors were temporarily assigned to supervise the night-shift employees. Most emphasis was placed on increasing the output of bags to meet the ever-increasing demand. It was suspected that only occasional reminders were made to double-check the bag weight-feeder. (A double-check is performed by systemati-. cally weighing a bag on a scale to determine whether the proper weight is being loaded by the weight-feeder. If there is significant deviation from 50 pounds, corrective adjustments are made to the weight-release mechanism.) To verify this expectation, the quality control staff randomly sampled the bag output and prepared the chart found on the previous page. Six bags were sampled and weighed each hour. Discussion Questions 1. What is your analysis of the bag weight problem? 2. What procedures would you recommend to maintain proper quality control? Source: Professor Jerry Kinard, Western Carolina University.. Case Study Morristown Daily Tribune In July 1998, the Morristown Daily Tribune published its first newspaper in direct competition with two other newspapers, the Morristown Daily Ledger and the Clarion Herald, a weekly publication. Presently, the Ledger is the most widely read newspaper in the area, with a total circulation of 38,500. The Tribune, however, has made significant inroads into the readership market since its inception. Total circulation of the Tribune now exceeds 27,000. Rita Bornstein, editor of the Tribune, attributes the success of the newspaper to the accuracy of its contents, a strong editorial section, and the proper blending of local, regional, national, and international news items. In addition, the paper has been successful in getting the accounts of several major retailers who advertise extensively in the display section. Finally, experienced reporters, photographers, copy writers,. PARAGRAPHS WITH ERRORS SAMPLE IN THE SAMPLE. FRACTION OF PARAGRAPHS WITH ERRORS (PER 100). typesetters, editors, and other personnel have formed a team dedicated to providing the most timely and accurate reporting of news in the area. Of critical importance to good-quality newspaper printing is accurate typesetting. To assure quality in the final print, Ms. Bornstein has decided to develop a procedure for monitoring the performance of typesetters over a period of time. Such a procedure involves sampling output, establishing control limits, comparing the Tribune’s accuracy with that of the industry, and occasionally updating the information. First, Ms. Bornstein randomly selected 30 newspapers published during the preceding 12 months. From each paper, 100 paragraphs were randomly chosen and were read for accuracy. The number of paragraphs with errors in each paper was recorded, and the fraction of paragraphs with errors in each sample was determined. The following table shows the results of the sampling:. SAMPLE. PARAGRAPHS WITH ERRORS IN THE SAMPLE. FRACTION OF PARAGRAPHS WITH ERRORS (PER 100). 1. 2. 0.02. 16. 2. 0.02. 2. 4. 0.04. 17. 3. 0.03. 3. 10. 0.10. 18. 7. 0.07. 4. 4. 0.04. 19. 3. 0.03. 5. 1. 0.01. 20. 2. 0.02. 6. 1. 0.01. 21. 3. 0.03. (table continued on page M1-24).

(25) M1-24. Module 1. STATISTICAL QUALITY CONTROL. PARAGRAPHS WITH ERRORS IN THE SAMPLE. FRACTION OF PARAGRAPHS WITH ERRORS (PER 100). SAMPLE. PARAGRAPHS WITH ERRORS IN THE SAMPLE. FRACTION OF PARAGRAPHS WITH ERRORS (PER 100). 7. 13. 0.13. 22. 7. 0.07. 8. 9. 0.09. 23. 4. 0.04. 9. 11. 0.11. 24. 3. 0.03. 10. 0. 0.00. 25. 2. 0.02. 11. 3. 0.03. 26. 2. 0.02. 12. 4. 0.04. 27. 0. 0.00. 13. 2. 0.02. 28. 1. 0.01. 14. 2. 0.02. 29. 3. 0.03. 15. 8. 0.08. 30. 4. 0.04. SAMPLE. Discussion Questions 1. Plot the overall fraction of errors ( p) and the upper and lower control limits on a control chart using a 95% confidence level. 2. Assume that the industry upper and lower control limits are 0.1000 and 0.0400, respectively. Plot them on the control chart.. 3. Plot the fraction of errors in each sample. Do all fall within the firm’s control limits? When one falls outside the control limits, what should be done?. Source: Professor Jerry Kinard, Western Carolina University.. Bibliography Berry, L. L., A. Parasuraman, and V. A. Zeithaml. “Improving Service Quality in America: Lessons Learned,” The Academy of Management Executive 8, 2 (May 1994): 32–52.. DeVor, R. E., T. Chang, and J. W. Sutherland. Statistical Quality Design and Control: Contemporary Concepts and Methods. New York: Macmillan Publishing Co., Inc., 1992.. Besterfield, D. H. Quality Control, 4th ed. Upper Saddle River, NJ: Prentice Hall, 1994.. Dobyns, L., and C. Crawford-Mason. Quality or Else: The Revolution in World Business. New York: Houghton Mifflin Company, 1991.. Carr, L. P. “Applying Cost of Quality to a Service Business,” Sloan Management Review 33, 4 (Summer 1992): 72. Costin, H. Readings in Total Quality Management. New York: Dryden Press, 1994.. Elsayed, E. A., and D. Dietrich. “Quality Control and Its Applications in Production Systems,” Industrial Engineering Research & Development 24, 5 (November 1992): 2–3.. Crosby, P. B. Let’s Talk Quality. New York: McGraw-Hill Book Company, 1989.. Feigenbaum, A. V. Total Quality Control, 3rd ed. New York: McGraw-Hill Book Company, 1991.. Crosby, P. B. Quality Is Free. New York: McGrawHill Book Company, 1979.. Juran, J. M. “Made in the U.S.A.: A Renaissance in Quality,” Harvard Business Review 14, 4 (July–August 1993): 35–38.. Deming, W. E. Out of the Crisis. Cambridge, MA: MIT Center for Advanced Engineering Study, 1986. Denton, D. K. “Lessons on Competitiveness: Motorola’s Approach,” Production and Inventory Management Journal 32, 3 (Third Quarter 1991): 22.. Mitra, A. Fundamentals of Quality Control and Improvement, 2nd ed. Upper Saddle River, NJ: Prentice Hall, 1998. Wheeler, D. J. “Why Three Sigma Limits?” Quality Digest (August 1996): 63–64..

(26) Appendix M1.1: Using QM for Windows for SPC. M1-25. ● APPENDIX M1.1: USING QM FOR WINDOWS FOR SPC QM for Windows’ quality control module has the ability to compute all of the SPC control charts that we introduced in this chapter. To illustrate, Program M1.2 uses the p-chart data for ARCO found in Section M1.5. It computes p-bar, the standard deviation, and upper and lower control limits. PROGRAM M1.2 QM for Windows Analysis of ARCO’s Data to Compute p-chart Control Limits.

(27) M O D U L E. 2. Dynamic Programming. LEARNING OBJECTIVES After completing this module, students will be able to: 1. 2. 3. 4. 5.. Understand the overall approach of dynamic programming. Use dynamic programming to solve the shortest-route problem. Develop dynamic programming stages. Describe important dynamic programming terminology. Describe the use of dynamic programming in solving knapsack problems.. MODULE OUTLINE M2.1 M2.2 M2.3 M2.4 M2.5. Introduction Shortest-Route Problem Solved by Dynamic Programming Dynamic Programming Terminology Dynamic Programming Notation Knapsack Problem Summary • Glossary • Key Equations • Solved Problem • Self-Test • Discussion Questions and Problems • Case Study: United Trucking • Internet Case Study • Bibliography. M2-1.

(28) M2-2. Module 2. DYNAMIC PROGRAMMING. M2.1 INTRODUCTION ●. Dynamic programming breaks a difficult problem into subproblems.. Dynamic programming is a quantitative analysis technique that has been applied to large, complex problems that have sequences of decisions to be made. Dynamic programming divides problems into a number of decision stages; the outcome of a decision at one stage affects the decision at each of the next stages. The technique is useful in a large number of multiperiod business problems, such as smoothing production employment, allocating capital funds, allocating salespeople to marketing areas, and evaluating investment opportunities. Dynamic programming differs from linear programming in two ways. First, there is no algorithm (like the simplex method) that can be programmed to solve all problems. Dynamic programming is, instead, a technique that allows us to break up difficult problems into a sequence of easier subproblems, which are then evaluated by stages. Second, linear programming is a method that gives single-stage (one time period) solutions. Dynamic programming has the power to determine the optimal solution over a one-year time horizon by breaking the problem into 12 smaller one-month time horizon problems and to solve each of these optimally. Hence it uses a multistage approach. Solving problems with dynamic programming involves four steps: Four Steps of Dynamic Programming 1. Divide the original problem into subproblems called stages. 2. Solve the last stage of the problem for all possible conditions or states. 3. Working backward from the last stage, solve each intermediate stage. This is done by determining optimal policies from that stage to the end of the problem (last stage). 4. Obtain the optimal solution for the original problem by solving all stages sequentially.. In this module we show you how to solve two types of dynamic programming problems: network and nonnetwork. The shortest-route problem is a network problem that can be solved by dynamic programming. The knapsack problem is an example of a nonnetwork problem that can be solved using dynamic programming.. M2.2 SHORTEST-ROUTE PROBLEM SOLVED BY DYNAMIC ● PROGRAMMING. The first step is to divide the problem into subproblems or stages.. George Yates is about to make a trip from Rice, Georgia (1) to Dixieville, Georgia (7). George would like to find the shortest route. Unfortunately, there are a number of small towns between Rice and Dixieville. His road map is shown in Figure M2.1. The circles on the map, called nodes, represent cities such as Rice, Dixieville, Brown, and so on. The arrows, called arcs, represent highways between the cities. The distance in miles is indicated along each arc. This problem can, of course, be solved by inspection. But seeing how dynamic programming can be used on this simple problem will teach you how to solve larger and more complex problems. Step 1: The first step is to divide the problem into subproblems or stages. Figure M2.2 reveals the stages of this problem. In dynamic programming, we usually start with the last part of the problem, stage 1, and work backward to the beginning of the problem or network, which is stage 3 in this problem. Table M2.1 summarizes the arcs and arc distances for each stage..

(29) FIGURE M2.1 Highway Map between Rice and Dixieville Lakecity. Athens 10 Miles. 4. 5 14. les. 1. ile s. 4. i 2M. Mi. 7 Dixieville. 4. 5 Miles. M. Brown. 1. Mi. les. les. 3. M. 6M. ile. ile. s. 2. 2M. ile s. Rice. s. 10 Miles. 2 Hope. 6 Georgetown. FIGURE M2.2 Three Stages for the George Yates Problem. A Node A Branch. 5 14. 4. 1. 10. 4. 5 2. 3. 12 6. 7 2. 4. 6. 10. 2 Stage 3. Stage 2. TABLE M2.1. Stage 1. Distance Along Each Arc. STAGE. ARC. ARC DISTANCE. 1. 5–7. 14. 6–7. 2. 4–5. 10. 3–5. 12. 3–6. 6. 2–5. 4. 2–6. 10. 1–4. 4. 1–3. 5. 1–2. 2. 2. 3. M2-3.

(30) M2-4. Module 2. DYNAMIC PROGRAMMING. FIGURE M2.3 Solution for the One-Stage Problem. 5 14. 4 1. Minimum Distance to Node 7 from Node 5. 14. 10. 4. 12 3 6. 5 2. 7 2. 4. 10. 6. 2 2. The second step is to solve the last stage—stage 1.. Minimum Distance to Node 7 from Node 6. Step 2: We next solve stage 1, the last part of the network. Usually, this is trivial. We find the shortest path to the end of the network, node 7 in this problem. At stage 1, the shortest paths from node 5, and node 6 to node 7 are the only paths. You may also note in Figure M2.3 that the minimum distances are enclosed in boxes by the entering nodes to stage 1, node 5 and node 6. The objective is to find the shortest distance to node 7. The following table summarizes this procedure for stage 1. As mentioned previously, the shortest distance is the only distance at stage 1. STAGE 1. Step 3 involves moving backward solving intermediate stages.. BEGINNING NODE. SHORTEST DISTANCE TO NODE 7. ARCS ALONG THIS PATH. 5. 14. 5–7. 6. 2. 6–7. Step 3: Moving backward, we now solve for stages 2 and 3. At stage 2 we will use Figure M2.4. If we are at node 4, the shortest and only route to node 7 is arcs 4–5 and 5–7. At node 3, the shortest route is arcs 3–6 and 6–7 with a total minimum distance of 8 miles. If we are at node 2, the shortest route is arcs 2–6 and 6–7 with a minimum total distance of 12 miles. This information is summarized in the stage 2 table: STAGE 2 BEGINNING NODE. SHORTEST DISTANCE TO NODE 7. ARCS ALONG THIS PATH. 4. 24. 4–5 5–7. 3. 8. 3–6 6–7. 2. 12. 2–6 6–7.

(31) M2.3 Dynamic Programming Terminology. Minimum Distance to Node 7 from Node 4. M2-5. FIGURE M2.4 Solution for the Two-Stage Problem. 24. 4 10 Minimum Distance to Node 7 from Node 3. 14. 8 12. 4. 1. 5 2. 3. 5. 14. 7. 6 2 4. 10. 6. 2 2 Minimum Distance to Node 7 from Node 2. 12. The solution to stage 3 can be completed using the accompanying table and the network in Figure M2.5. STAGE 3 BEGINNING NODE. SHORTEST DISTANCE TO NODE 7. ARCS ALONG THIS PATH. 1. 13. 1–3 3–6 6–7. Step 4: To obtain the optimal solution at any stage, all we consider are the arcs to the next stage and the optimal solution at the next stage. For stage 3, we only have to consider the three arcs to stage 2 (1–2, 1–3, and 1–4) and the optimal policies at stage 2, given in a previous table. This is how we arrived at the preceding solution. When the procedure is understood, we can perform all the calculations on one network. You may want to study the relationship between the networks and the tables because more complex problems are usually solved by using tables only.. M2.3 DYNAMIC PROGRAMMING TERMINOLOGY ● Regardless of the type or size of a dynamic programming problem, there are some important terms and concepts that are inherent in every problem. Some of the more important follow: 1. Stage: a period or a logical subproblem. 2. State variables: possible beginning situations or conditions of a stage. These have also been called the input variables.. The fourth and final step is to find the optimal solution after all stages have been solved..

(32) M2-6. Module 2. DYNAMIC PROGRAMMING. FIGURE M2.5 Solution for the Three-Stage Problem. 24 Minimum Distance to Node 7 from Node 1. 4. 13. 8. 10 14. 5 14. 4 1. 5 2. 12 3 6. 7 2. 4. 10. 6. 2 2 12. 3. Decision variables: alternatives or possible decisions that exist at each stage. 4. Decision criterion: a statement concerning the objective of the problem. 5. Optimal policy: a set of decision rules, developed as a result of the decision criteria, that gives optimal decisions for any entering condition at any stage. 6. Transformation: normally, an algebraic statement that reveals the relationship between stages.. IN ACTION. Dynamic Programming in Nursery Production Decisions. Managing a nursery that produces ornamental plants is difficult. In most cases, ornamental plants increase in value with increased growth. This value-added growth makes it difficult to determine when to harvest the plants and place them on the market. When plants are marketed earlier, revenues are generated earlier and the costs associated with plant growth are minimized. On the other hand, delaying the harvesting of the ornamental plants usually results in higher prices. But are the additional months of growth and costs worth the delay? In this case, dynamic programming was used to determine the optimal growth stages for ornamental plants. Each stage was associated with a possible growth level. The state variables included acres of production of ornamental plants and carryover plants from previous growing seasons. The ob-. jective of the dynamic programming problem was to maximize the after-tax cash flow. The taxes included self-employment, federal income, earned income credit, and state income taxes. The solution was to produce one- and three-gallon containers of ornamental plants. The one-gallon containers are sold in the fall and carried over for spring sales. Any one-gallon containers not sold in the spring are combined into threegallon container products for sale during the next season. Using dynamic programming helps to determine when to harvest to increase after-tax cash flow. Source: Stokes, Jeffery et al. “Optimal Marketing of Nursery Crops From Container-Based Production Systems,” American Journal of Agricultural Economics (February 1997): 235..

(33) M2.4 Dynamic Programming Notation. M2-7. In the shortest-route problem, the following transformation can be given: distance from the beginning of a given stage to the last node. . distance from the beginning of the previous stage to the last node. distance from the given stage to the previous stage. This relationship shows how we were able to go from one stage to the next in solving for the optimal solution to the shortest-route problem. In more complex problems, we can use symbols to show the relationship between stages. State variables, decision variables, the decision criterion, and the optimal policy can be determined for any stage of a dynamic programming problem. This is done here for stage 2 of the George Yates shortest-route problem. 1. State variables for stage 2 are the entering nodes, which are (a) Node 2 (b) Node 3 (c) Node 4 2. Decision variables for stage 2 are the following arcs or routes: (a) 4–5 (b) 3–5 (c) 3–6 (d) 2–5 (e) 2–6 3. The decision criterion is the minimization of the total distances traveled. 4. The optimal policy for any beginning condition is shown in Figure M2.6 and following the table below. GIVEN THIS ENTERING CONDITION. THIS ARC WILL MINIMIZE TOTAL DISTANCE TO NODE 7. 2. 2–6. 3. 3–6. 4. 4–5. Figure M2.6 may also be helpful in understanding some of the terminology used in the discussion of dynamic programming.. M2.4 DYNAMIC PROGRAMMING NOTATION ● In addition to dynamic programming terminology, we can also use mathematical notation to describe any dynamic programming problem. This helps us to set up and solve the problem. Consider stage 2 in the George Yates dynamic programming problem first discussed in Section M2.2. This stage can be represented by the diagram shown in Figure M2.7 (as could any given stage of a given dynamic programming problem). As you can see, for every stage, we have an input, decision, output, and return. Look again at stage 2 for the George Yates problem in Figure M2.6. The input to this stage is s2, which consists of nodes 2, 3, and 4. The decision at stage 2, or choosing which arc will. An input, decision, output, and return are specified for each stage..

(34) M2-8. Module 2. DYNAMIC PROGRAMMING. FIGURE M2.6 Stage 2 from the ShortestRoute Problem. State variables are the entering nodes.. 24. 10. 4. 14 Decision variables are all the arcs.. 8 5 12 3 6. 1. 4. 7 6. 10. 2 2 The optimal policy is the arc, for any entering node, that will minimize total distance to the destination at this stage.. 12. Stage 2. lead to stage 1, is represented by d2. The possible arcs or decisions are 4–5, 3–5, 3–6, and 2–6. The output to stage 2 becomes the input to stage 1. The output from stage 2 is s1. The possible outputs from stage 2 are the exiting nodes, nodes 5 and 6. Finally, each stage has a return. For stage 2, the return is represented by r2. In our shortest-route problem, the return is the distance along the arcs in stage 2. These distances are 10 miles for arc 4–5, 12 FIGURE M2.7 Input, Decision, Output, and Return for Stage 2 in George Yates’s Problem. Decision d2. Stage 2 Input s2. Output s1. Return r2.

(35) M2.5 Knapsack Problem. M2-9. miles for arc 3–5, 6 miles for arc 3–6, and 10 miles for arc 2–6. The same notation applies for the other stages and can be used at any stage. In general, we will use the following notation for these important concepts: sn input to stage n. (M2-1). dn decision at stage n. (M2-2). rn return at stage n. (M2-3). You should also note that the input to one stage is also the output from another stage. For example, the input to stage 2, s2, is also the output from stage 3 (see Figure M2.7). This leads us to the following equation: sn 1 output from stage n. (M2-4). The final concept is transformation. The transformation function allows us to go from one stage to another. The transformation function for stage 2, t2, converts the input to stage 2, s2, and the decision made at stage 2, d2, to the output from stage 2, s1. Because the transformation function depends on the input and decision at any stage, it can be represented as t2 (s2, d2). In general, the transformation function can be represented as follows: tn transformation function at stage n. The input to one stage is the output from another stage.. A transformation function allows us to go from one stage to another.. (M2-5). The following general formula allows us to go from one stage to another using the transformation function: sn1 tn (sn , dn). (M2-6). Although this equation may seem complex, it is really just a mathematical statement of the fact that the output from a stage is a function of the input to the stage and any decisions made at that stage. In the George Yates shortest-route problem, the transformation function consisted of a number of tables. These tables showed how we could progress from one stage to another in order to solve the problem. For more complex problems, we need to use dynamic programming notation instead of tables. Another useful quantity is the total return at any stage. The total return allows us to keep track of the total profit or costs at each stage as we solve the dynamic programming problem. It can be given as follows: fn total return at stage n. (M2-7). M2.5 KNAPSACK PROBLEM ● The knapsack problem involves the maximization or the minimization of a value, such as profits or costs. Like a linear programming problem, there are restrictions. Imagine a knapsack or pouch that can only hold a certain weight or volume. We can place different types of items in the knapsack. Our objective will be to place items in the knapsack to maximize total value without breaking the knapsack because of too much weight or a similar restriction.. Types of Knapsack Problems There are many kinds of problems that can be classified as knapsack problems. Choosing items to place in the cargo compartment of an airplane and selecting which payloads to put on the next NASA Space Shuttle are examples. The restriction can be volume, weight,. The total return function allows us to keep track of profits and costs..

(36) M2-10. Module 2. DYNAMIC PROGRAMMING. or both. Some scheduling problems are also knapsack problems. For example, we may want to determine which jobs to complete in the next two weeks. The two-week period is the knapsack, and we want to load it with jobs in such a way as to maximize profits or minimize costs. The restriction is the number of days or hours during the two-week period.. Roller’s Air Transport Service Problem The objective of this problem is to maximize profits.. Rob Roller owns and operates Roller’s Air Transport Service, which ships cargo by plane to most large cities in the United States and Canada. The remaining capacity for one of the flights from Seattle to Vancouver is 10 tons. There are four different items that Rob can ship between Seattle and Vancouver. Each item has a weight in tons, a net profit in thousands of dollars, and a total number of that item that is available for shipping. This information is presented in Table M2.2.. MODELING IN THE REAL WORLD Defining the Problem. Developing a Model. Reducing Electric Production Costs Using Dynamic Programming. The Southern Company, with service areas in Georgia, Alabama, Mississippi, and Florida, is a major provider of electric service, with about 240 generating units. In recent years, fuel costs have increased faster than other costs. Annual fuel costs are about $2.5 billion, representing about one-third of total expenses for the Southern Company. The problem for the Southern Company is to reduce total fuel costs. To deal with this fuel cost problem, the company developed a state-of-the-art dynamic programming model. The dynamic programming model is embedded in the Wescouger optimization program, which is a computer program used to control electric generating units and reduce fuel costs through better utilization of existing equipment.. Acquiring Input Data. Data were collected on past and projected electric usage. In addition, daily load/generation data were analyzed. Load/generation charts were used to investigate the fuel requirements for coal, nuclear, hydroelectric, and gas/oil.. Developing a Solution. The solution of the dynamic programming model provides both short-term planning guidelines and long-term fuel usage for the various generating units. Optimal maintenance schedules for generating units are obtained using Wescouger.. Testing the Solution. Analyzing the Results. Implementing the Results. To test the accuracy of the Wescouger optimization program, Southern used a real-time economic dispatch program. The results were a very close match. In addition, the company put the solution through an acid test, in which seasoned operators compared the results against their intuitive judgment. Again, the results were consistent. The results were analyzed in terms of their impact on the use of various fuels, the usage of various generating units, and maintenance schedules for generating units. Analyzing the results also revealed other needs. This resulted in a full-color screen editing routine, auxiliary programs to automate data input, and software to generate postanalysis reports. The Southern Company implemented the dynamic programming solution. Over a seven-year period, the results saved the company over $140 million.. Source: S. Erwin, et al. “Using an Optimization Software to Lower Overall Electric Production Costs for Southern Company,” Interfaces 21, 1 (January–February 1991): 27–41..

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