2Control Charts for Measurements of Quality Example Usage: number of ounces per bottle; diameters of ball bearings; lengths of screwsMean (x-bar) chartsTracks the central tendency (the average or mean value observed) over timeRange (R) charts:Tracks the spread of the distribution (largest - smallest) over time
3X-Bar Chart Computations First, find “xbar-bar”, the average of the averages2. Now find , where σ is the standard deviation and n is thesize of each sample3). Find the upper control limit (UCL) and lower control limit (LCL) using the following formulas:Number of sample averagesZ is the number of sigma limits specified in the problem. For “3 sigma limits” use z = 3, for example.
4Assume the standard deviation of the process is given as 1 Assume the standard deviation of the process is given as 1.13 ounces Management wants a 3-sigma chart (only 0.26% chance of alpha error) Observed values shown in the table are in ounces. Calculate the UCL and LCL.Sample 1Sample 2Sample 3Observation 115.816.116.0Observation 215.9Observation 3Observation 4Sample means15.87515.975
5Range or R Chart k = # of sample ranges A range chart measures the variability of the process using the averageof the sample ranges (range = largest – smallest)The values of D3 and D4 are special constants whosevalues depend on the sample size. These constants will be given to youin a chart.
6Range Chart FactorsD3D420.003.2732.5742.2852.1162.0070.081.9280.141.8690.181.82100.221.78110.261.74120.281.72130.311.69140.331.67150.351.65Factors for R-ChartSample Size (n)
8Ten samples of 5 observations each have been taken form a Soft drink bottling plant in order to test for volume dispersionin the bottling process. The average sample range was foundTo be .5 ounces. Develop control limits for the sample range.
9P Fraction Defective Chart P-ChartsP Fraction Defective Chart“Proportion charts”Used for yes-or-no type judgments (acceptable/not acceptable, works/doesn’t work, on time/late, etc.)p = proportion of nonconforming itemsControl limits are based on= average proportion of nonconforming items
10P-Chart Computations 1). Find p-bar: 2). Compute 3). Compute UCL and LCL using the formulas:Number of observations per sampleAs with the X-Bar chart, z is the number of sigma limits specified in the problemIf LCL turns out to be negative, set it to 0 (lower limit can’t be negative—why?)
11P-Chart Example: A Production manager for a tire company has inspected the number of defective tires in five random samples with 20 tires in each sample. The table below shows the number of defective tires in each sample of 20 tires. Z= 3. Calculate the control limits.SampleNumber of Defective TiresNumber of Tires in each SampleProportion Defective1320.152.10.0545Total9100.09
12C-Charts “Count charts” Used when looking at # of defects Control limits are based on average number of defects,
13Number-of-Defectives or C Chart C-Chart ComputationsNumber-of-Defectives or C Chart1). Compute c-bar:2). Compute3). Compute LCL and UCL using the formulas:As with the X-Bar chart, z is the number of sigma limits specified in the problemAs with the P-Bar chart, if the LCL turns out to be negative, set LCL to 0 (LCL can’t be negative, why?
14C-Chart Example: The number of weekly customer complaints are monitored in a large hotel using a c-chart. Develop three sigma control limits using the data table below. Z=3.WeekNumber of Complaints13245678910Total22
15Process Capability“Capability” : Can a process or system meet its requirements?Cp < 1: process not capable of meeting design specsCp ≥ 1: process capable of meeting design specsCp assumes that the process is centered on the specification range, which may not be the case!To see if a process is centered, we use Cpk:
16Cpk Cp=Cpk when process is centered min = “minimum of the two” = mean of the processA value of Cpk < 1 indicates that the process is notcentered.Cp=Cpk when process is centered
17ExampleDesign specifications call for a target value of /-0.2 ounces. Observed process output has a mean of 15.9 and a standard deviation of 0.1 ounces. Is the process capable?LSL = = 15.8USL = = 16.2
19Critical Path Method (CPM) CPM is an approach to scheduling and controlling project activities.The critical path: Longest path through the processRule 1: EF = ES + Time to complete activityRule 2: the ES time for an activity equals the largest EF time of all immediate predecessors.Rule 3: LS = LF – Time to complete activityRule 4: the LF time for an activity is the smallest LS of all immediate successors.
23Step 3: Identify All Unique Paths And Path Durations Path Duration = Sum of all task times along the pathPathDurationABDEGHJK40ABDEGIJK41ACFGHJK22ACFGIJK23Critical path
24Adding Feeder Buffers to Critical Chains The theory of constraints, the basis for critical chains, focuses on keeping bottlenecks busy.Time buffers can be put between bottlenecks in the critical pathThese feeder buffers protect the critical path from delays in non-critical paths
26Some Network Definitions All activities on the critical path have zero slackSlack defines how long non-critical activities can be delayed without delaying the projectSlack = the activity’s late finish minus its early finish (or its late start minus its early start)Earliest Start (ES) = the earliest finish of the immediately preceding activityEarliest Finish (EF) = is the ES plus the activity timeLatest Start (LS) and Latest Finish (LF) depend on whether or not the activity is on the critical path
27B(6) D(6) A(4) C(3) F(5) E(14) G(2) I(3) H(2) J(4) K(2) ES=0 EF=4 LS=0 LF=4ES=4+6=10EF=10LS=4LF=10ES=10EF=16ES=16EF=30ES=32EF=34ES=35EF=39ES=39EF=41EF=35LS=32LF=35ES=30EF=32LS=30LF=32ES=7EF=12LS=25LF=30ES=4EF=7LS=22LF=25Calculate EarlyStarts & FinishesLatest EF= Next ESStrategy: Find all the ES’s and EF’s first by moving left to right (start to finish).Then find LF and LS by working backward (finish to start)
29Activity Slack = TLS - TES = TLF - TEF Activity Slack TimeTES = earliest start time for activityTLS = latest start time for activityTEF = earliest finish time for activityTLF = latest finish time for activityActivity Slack = TLS - TES = TLF - TEFIf an item is on the critical path, there is no slack!!!!
30Calculate Activity Slack The critical path was ABDEGIJKNotice that the slack for these task times is 0.
32Arrival & Service Patterns Arrival rate:The average number of customers arriving per time periodService rate:The average number of customers that can be serviced during the same period of timeArrival rate and service rate must be in the same units!!
33Infinite Population, Single-Server, Single Line, Single Phase Formulae
34Infinite Population, Single-Server, Single Line, Single Phase Formulae
35ExampleA help desk in the computer lab serves students on a first-come, first served basis. On average, 15 students need help every hour. The help desk can serve an average of 20 students per hour.Based on this description, we know:µ = 20λ= 15Note that both arrival rate and service rate are in hours, so we don’t need to do any conversion.
36Average Number of Students in the System Average UtilizationAverage Number of Students in the SystemAverage Number of Students Waiting in Line
37Average Time a Student Spends in the System .2 hours or 12 minutes
38Average Time a Student Spends Waiting (Before Service) Too long?After 5 minutes peopleget anxious
39Suppose that customers arrive according to a Poisson distribution at an average rate of 60 per hour, and the average (exponentially distributed) service time is 45 seconds per customer. What is the average number of customers in the system?Convert to hours first!