_Solution__Quality_Control_HW

Quality Control HW Solutions Problem 1: The management of the Diners Delight restaurant chain is in the process of establishing quality control charts for the time that its service people give to each customer. Management thinks the length of time that each customer is given should remain within certain limits to enhance service quality. For simplicity, the data from four samples are displayed below. In each sample, six service people were randomly selected, and the customer service they provided was observed. The activities that the service people were performing were identified, and the time to service each customer was recorded as below. Service time (in seconds) Service person Sample 1 Sample 2 Sample 3 Sample 4 1 200 150 175 90 2 120 85 105 75 3 83 93 130 150 4 68 150 145 175 5 110 90 75 105 6 115 65 115 125 116 105.5 124.17 120 R 200-68=132 150-65=85 175-75=100 175-75=100 Use the given information, please (a) Find the 99.73% (3-sigma) upper and lower control limits for X-bar chart and R-chart. (b) After the control chart was established, a sample of six service personnel was observed, and the following customer service times in seconds were recorded: 180, 125, 110, 98, 156 and 190. Is the process in control? Any corrective action needed? Solution: (a) X =( 116 + 105.5 + 124.17 + 120 )/ 4 = 116.42 R =( 132 + 85 + 100 + 100 )/ 4 = 104.25 Sample size n = 6  A2 = 0.48, D3=0, and D4=2.0 Control limits for mean: UCL = ¯ ¯ X + A 2 ¯ R = 116 .42 +( .48 )( 104.25 )= 166.46 1

LCL = ¯ ¯ X − A 2 ¯ R = 116 .42 −( .48 )( 104.25 )= 66.38 Control limits for range: UCL = D 4 ¯ R =( 2.0 )( 104.25 )= 208.5 LCL = D 3 ¯ R =( 0 )( 104.25 )= 0 (b) Sample mean: = (180 + 125 + 110 + 98 + 156 + 190)/6 = 143.17 Sample range: R = 190 - 98 = 92 and R are both within the control limits. Process is in control, no action is needed. Problem 2: Several complaints recently have been sent to the Logan police department regarding the increasing incidence of congestion on the city’s streets. The complaints attribute the cause of these traffic tie- ups to a lack of synchronization of the traffic lights. The lights are controlled by a main computer system, and adjusting this program is costly. Therefore, the controllers are reluctant to change the situation unless a clear need is shown. During the past year, the police department has collected 1,000 data each month at major intersections. The data were as shown below. Month Congestion incidence January 14 February 18 March 14 April 12 May 16 June 8 July 19 August 12 September 14 October 7 November 10 December 18 (a) Construct a control chart with 3-sigma quality standard based on the above data. 2

Quality Control Reference I. Factors for 3  Control Limits for ¯ x and R Charts Factors For R Charts Number of Observations in a sample, n Factor for ¯ x Chart, A 2 Lower Control Limit, D 3 Upper Control Limit, D 4 2 1.88 0 3.27 3 1.02 0 2.57 4 0.73 0 2.28 5 0.58 0 2.11 6 0.48 0 2.00 7 0.42 0.08 1.92 8 0.37 0.14 1.86 9 0.34 0.18 1.82 10 0.31 0.22 1.78 11 0.29 0.26 1.74 12 0.27 0.28 1.72 13 0.25 0.31 1.69 14 0.24 0.33 1.67 15 0.22 0.35 1.65 16 0.21 0.36 1.64 17 0.20 0.38 1.62 18 0.19 0.39 1.61 19 0.19 0.40 1.60 20 0.18 0.41 1.59 II. Quality Control formulas Name Formula Control Charts ¯ x -chart ¯ ¯ x ± A 2 ¯ R R-chart UCL = D 4 ¯ R ; LCL = D 3 ¯ R p-chart ¯ p ± z √ ¯ p ( 1 −¯ p ) n ; ¯ p = total . defects / total ⋅ observations Process Capability C pk = min [ Upper . specification − ¯ x 3 σ , ¯ x − Lower . specification 3 σ ] 4