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Islamic University, Gaza - Palestine Islamic University, Gaza - Palestine Department of Industrial Engineering Department of Industrial Engineering Statistical Quality Control Process Capability Presented by Presented by Dr. Eng. Abed Schokry Dr. Eng. Abed Schokry Islamic University, Gaza - Palestine Islamic University, Gaza - Palestine Process Capability Process Capability • Process capability refers to the ability of a process to produce products or provide services capable of meeting the specifications set • Tolerances or specifications by the customer or designer. – Range of acceptable values established by engineering • Specification limits are set by management in response to customers’ design or customer requirements expectations • Process variability • The upper specification limit (USL) is the largest value that can be – Natural variability in a process obtained and still conform to customers’ expectations • Process capability • The lower specification limit (LSL) is the smallest value that is still conforming – Process variability relative to specification 1 Islamic University, Gaza - Palestine Islamic University, Gaza - Palestine Tolerance Limits vs. Process Capability Process Capability Specification Width PROCESS SPREAD SPECIFIED TOLERANCES ` SPECIFIED TOLERANCES ` PROCESS SPREAD Specification Width Actual Process Width A capable process An incapable process Actual Process Width Islamic University, Gaza - Palestine Islamic University, Gaza - Palestine Process Capability Specification limits • Specification limits, the allowable spread of the individuals, are • A process capability index is an aggregate measure of a compared with the spread of the process to determine how capable process’s ability to meet specification limits the process is of meeting the specifications. Three different situations can exist when specifications and are compared: • The larger the value, the more capable a process is of meeting requirements (I) The process spread can be less than the spread of the specification limits; (II) The process spread can be equal to the spread of the specification limits; (III) The process spread can be greater than the spread of the specification limits. 2 Islamic University, Gaza - Palestine Islamic University, Gaza - Palestine (I)The spread of the individuals is less than the spread of (II) The process spread can be equal to the spread of the specifications the specification limits • In this situation, is equal to the tolerance • The control limits have been placed on the diagram, as well as the spread of the process averages (dotted line). • As long as the process remains in control and centered, with no change in process variation, the parts produced will be within specification. • The spread of the process individuals is shown by the solid line. As expected, the spread of the individual values is greater than the spread of the averages; however, the values are still within the specification • A shift in the process mean will result in the production of parts that limits. are out of specification. An increase in the variation present in the process also creates an out-of-specification situation. • This allows for more room for process shifts while staying within the specifications. • Notice that even if the process drifts out of control, the change must be dramatic before the parts are considered out of specification. Islamic University, Gaza - Palestine Islamic University, Gaza - Palestine (III) The spread is greater than the tolerance spread The capability index Cp • Case III: The spread is greater than the tolerance spread. • The capability index Cp is the ratio of tolerance (USL – LSL) and 6s • Even though the process is exhibiting only natural patterns of variation, it is incapable of meeting the specifications set by the customer. USL - LSL C = • To correct this problem, management intervention will be necessary p s in order to change the process to decrease the variation. 6 • The capability of the process cannot be improved without changing the existing process. 3 Islamic University, Gaza - Palestine Islamic University, Gaza - Palestine Capability ratio C Capability ratio Cr pk • Cpk is the ratio that reflects how the process is performing in terms of a nominal, center, or target value: • Capability ratio Cr s 6 Z(min) C = C r USL - LSL pk 3 USL - X Z(USL) = where Z(min) is the smaller of s X - LSL or Z(LSL) = s Islamic University, Gaza - Palestine Islamic University, Gaza - Palestine The relationships between Cp and Cpk The relationships between Cp and Cpk (Cont.) 1. When Cp has a value of 1.0 or greater, the process is producing product 6. When Cpk has a value less than 1.00, it indicates the process is producing capable of meeting specifications. product that does not conform to specifications. 7. A Cp value of less than 1.00 indicates that the process is not capable. 2. The Cp value does not reflect process centering. 8. A Cpk value of zero indicates the process average is equal to one of the 3. When the process is centered, Cp = Cpk. specification limits. 4. Cpk is always less than or equal to Cp. 9. A negative Cpk value indicates that the average is outside the specification limits. 5. When Cp is greater than or equal to 1.0 and Cpk has a value of 1.00 or more, it indicates the process is producing product that conforms to specifications. 4 Islamic University, Gaza - Palestine Islamic University, Gaza - Palestine Meanings of Cpk Measures Limitations of Capability Indexes 1. Process may not be stable 2. Process output may not be normally distributed Cpk = negative number 3. Process not centered but Cp is used Cpk = zero http://elsmar.com/Cp_vs_Cpk.html Cpk = between 0 and 1 Cpk = 1 Cpk > 1 Islamic University, Gaza - Palestine Islamic University, Gaza - Palestine Estimating Process Capability Process Capability Ratio If the process is centered use Cp • Must first have an in-control process specification width Process capability ratio, Cp = • Estimate the percentage of product or service within process width specification upper specification – lower specification • Assume the population of X values is approximately normally Cp = 6 distributed with mean estimated by and standard deviation estimated by If the process is not centered use Cpk X X LTL UTL - X R / d C = min or 2 pk 3 3 5 Islamic University, Gaza - Palestine Islamic University, Gaza - Palestine Cp Index CPL and CPU • A measure of potential process performance is the Cp • To measure capability in terms of actual process index performance: X LSL USL LSL specification spread CPL C p (3 R d/ 2 ) (6 R / d2 ) process spread USL X CPU – Cp > 1 implies a process has the potential of having more than 99.73% of outcomes within specifications (3 R / d2 ) – CPL (CPU) > 1 implies that the process mean is more than 3 standard deviation away from the lower (upper) specification limit Islamic University, Gaza - Palestine Islamic University, Gaza - Palestine CPL and CPU Cpk Index • Used for one-sided specification limits • The most commonly used capability index is the Cpk index • Measures actual process performance for characteristics – Use CPU when a characteristic only has a USL with two-sided specification limits Cpk = min(CPL, CPU) – Use CPL when a characteristic only has an LSL – Cpk = 1 indicates that the process average is 3 standard deviation away from the closest specification limit – Larger Cpk indicates greater capability of meeting the requirements 6 Islamic University, Gaza - Palestine Islamic University, Gaza - Palestine Process Capability & Tolerance ProcessProcess CapabilityCapability Lower Upper •When spec. established without knowing whether process capable Specification Specification of meeting it or not serious situations can result •Process capable or not – actually looking at process spread, which A. Process variability is called process capability (6) matches specifications Lower Upper Specification Specification •Let’s define specification limit as tolerance (T) : T = USL –LSL •3 types of situation can result B. Process variability Lower Upper the value of 6 < USL-LSL well within specifications the value of 6 = USL - LSL Specification Specification the value of 6 > USL - LSL C. Process variability exceeds specifications Islamic University, Gaza - Palestine Islamic University, Gaza - Palestine 33SigmaSigma and 66SigmaSigma Quality Case I situation Lower Upper specification specification 1350 ppm 1350 ppm 1.7 ppm 1.7 ppm Process mean +/- 3 Sigma +/- 6 Sigma 7 Islamic University, Gaza - Palestine Islamic University, Gaza - Palestine Case II situation Case 3 situation Islamic University, Gaza - Palestine Islamic University, Gaza - Palestine Both Cp and Cpk are identical Process Capability (6) And Tolerance because process mean is at the center of the specification spread As the process mean starts to • Cp - Capability Index Formulas drift away from the center of the T = U-L specification spread, value of Cp = 1 Case II 6 = T Cp = (T)/6 Cpk starts getting smaller Cp > 1 Case I 6 < T Z(min) (although Cp does not change) Cp < 1 Case III 6 > T Cpk = Usually Cp = 1.33 (de facto std.) 3 • Measure of process performance USL x Z (USL) = • Shortfall of Cp - measure σ not in terms of nominal or target value >>> must use Cpk x LSL 8 Islamic University, Gaza - Palestine Islamic University, Gaza - Palestine Example Interpreting the Process Capability Index Determine Cp and Cpk for a Solution process with average 6.45, Cp= T/6= 0.2/6(0.03)=1.11 Cpk < 1 Not Capable = 0.030, having USL = Cpk = Z(min)/3 6.50 , LSL = 6.30 -- T = 0.2 Z(U) = (USL -x)/ = 6.50- Cpk > 1 Capable at 3 6.45)/0.03 = 1.67 L T U Z(L) = (x –LSL)/ = 6.45-6.30)/0.03 = 5.00 Cpk > 1.33 Capable at 4 Cpk = 1.67/3 = 0.56 Process NOT capable since not Cpk > 1.67 Capable at 5 centered. Cp > 1 doesn’t mean capable. Have to check Cpk 6.30 6.45 6.50 Cpk > 2 Capable at 6 x = Islamic University, Gaza - Palestine Islamic University, Gaza - Palestine Process Capability Example Process Capability: Hotel Example Solution You are the manager of a 500-room hotel.
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