Question

ferry services are getting more popular among tourists in Turkey. Apart from safety aspects, timing of the journeys is always monitored, as lengthier and shorter journeys than scheduled are not desirable. For a service between Istanbul and Bursa in the Sea of Marmara, the ferries’ travel times are monitored for 15 days, four sample journeys each day. The results are shown in the table below.

MEAN(MINUTES)	MINIMUM	MAXIMUM
121	117	125
123	119	126
124	119	125
115	113	117
114	111	118
126	118	130
130	122	133
122	119	125
123	118	124
119	117	125
128	118	133
124	120	127
116	112	120
120	117	124
121	118	126

Value for A2, D3 and D4

Sample Size, n	A2	D3	D4
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

(i) Develop an X-bar chart and an R-chart. Show your workings clearly.

(ii) Plot the X-bar chart and R-chart. From the charts, what comments can you make about the process?

Answer 1

i) To find the control limits of x-bar chart we first need to find the means of all means (x-double bar) and the R-bar (mean of Range)

As all the sample means are given, x-double bar can be found as below

x-double bar = (121+123+124+115+114+126+130+122+123+119+128+124+116+120+121)/15 = 1826/15 = 121.73

As maximum and minimum for each sample is given, the range for each sample is simply the difference between maximum and minimum. Hence we have the sample ranges as below

Range for sample 1 = 125-117 = 8
Range for sample 2 = 126-119 = 7
Range for sample 3 = 125-119 = 6
Range for sample 4 = 117-113 = 4
Range for sample 5 = 118-111 = 7
Range for sample 6 = 130-118 = 12
Range for sample 7 = 133-122 = 11
Range for sample 8 = 125-119 = 6
Range for sample 9 = 124-118 = 6
Range for sample 10 = 125-117 = 8
Range for sample 11 = 133-118 = 15
Range for sample 12 = 127-120 = 7
Range for sample 13 = 120-112 = 8
Range for sample 14 = 124-117 = 7
Range for sample 15 = 126-118 = 8

R-bar is the average of above ranges. Hence R-bar is as below

R-bar = (8+7+6+4+7+12+11+6+6+8+15+7+8+7+8)/15 = 120/8 = 8

Now for the x-bar chart the control limits are given as per below formula

Control Limits = (x-double bar) A₂*(R-bar)

From the control chart tables given for the sample size of 4, A₂ = 0.73

The Central Line (CL) is simply the x-double bar. Hence the parameters for x-bar chart are as below

CLx = 121.73

UCL_x = x-double bar+A₂*(R-bar) = 121.73+0.73*8 = 127.57

LCL_x = x-double bar-A₂*(R-bar) = 121.73-0.73*8 = 115.89

For the range control chart the control limits are given as below

LCL = D₃*R-bar

UCL = D4*R-bar

The central line for this chart is simply the R-bar

For sample size of 4, we have D₃ = 0 and D₄ = 2.28. Hence the parameters for range control chart are

CL_R = 8

LCL_R = 0*8 = 0

UCLR = 2.28*8 = 18.24

b) The plots of x-bar and R-bar using the above parameters are as below

From the above charts we can see that while the range for the process is in control, the means are not in control, there are 4 samples which lie outside the control limits. Hence we can see that the process is not in control.