Question

You’ve just been promoted to a management position at Worcester Corset Candies Company. Congrats! Immediately, you get a call that Finance noticed a cost variance in your department. It appears that you are consuming too few ingredients for your scheduled output. A box of Special Mixed Corset Candies is supposed to be 140 ounces, and your annual bonus is paid on how close you manage your department’s per/unit costs. To begin analyzing the process, you ask your fill machine operator to record the weights of 9 candy cartons each hour for the next 11 hours. (HINT: Save yourself a lot of time and do this in Excel). Are all points on the R-chart are within the control limits? Are all points on the X-bar chart are within the control limits? So, manager, is your candy making process considered in control?

Data provided

Batch	Weight 1	Weight 2	Weight 3	Weight 4	Weight 5	Weight 6	Weight 7	Weight 8	Weight 9
1	147	136	138	138	131	130	134	148	144
2	148	137	149	134	134	143	144	150	140
3	138	141	130	140	135	132	138	134	133
4	141	143	149	133	143	149	132	135	130
5	135	141	140	139	135	150	136	130	135
6	133	146	142	136	150	140	144	147	150
7	138	137	145	138	134	149	130	131	147
8	144	137	133	149	131	141	143	149	146
9	145	134	149	140	142	130	150	142	148
10	135	146	130	141	135	137	141	140	150
11	149	142	149	143	142	137	134	139	140

Answer 1

Calculating the x bar and Range of the data

Batch	Weight 1	Weight 2	Weight 3	Weight 4	Weight 5	Weight 6	Weight 7	Weight 8	Weight 9	Mean	Range
1	147	136	138	138	131	130	134	148	144	138.44	18
2	148	137	149	134	134	143	144	150	140	142.11	16
3	138	141	130	140	135	132	138	134	133	135.67	11
4	141	143	149	133	143	149	132	135	130	139.44	19
5	135	141	140	139	135	150	136	130	135	137.89	20
6	133	146	142	136	150	140	144	147	150	143.11	17
7	138	137	145	138	134	149	130	131	147	138.78	19
8	144	137	133	149	131	141	143	149	146	141.44	18
9	145	134	149	140	142	130	150	142	148	142.22	20
10	135	146	130	141	135	137	141	140	150	139.44	20
11	149	142	149	143	142	137	134	139	140	141.67	15

Mean of each Batch = Average of Weight 1 to weight 9

Range of each batch = Max of values between Weight 1 to weight 9 - Min of values between Weight 1 to weight 9

x double bar = Mean of the averages of all batches from 1 to 11 = 140.02

Average range or R bar = Average of all ranges calculated from batch 1 to batch 11 = 17.55

Control Limits of X bar chart:

UCL = X double bar + A_2*R bar = 140.02+0.337*17.55 = 145.93

LCL = X double bar - A_2*R bar = 140.02-0.337*17.55 = 134.10

Control Limts of R bar chart:

UCL = D_4*R bar = 1.816*17.55 = 31.87

LCL = D_3*R bar = 0.184*17.55 = 3.23

A_2, D_3 and D_4 are the control chart constant. The value of these constanct can be found from the Contol Chart constant table.

Now plotting the X bar values (Mean of all batches). The number of batches on x axis and values on y axis.

The upper limit for x bar chart is 145.93 and lower limit is 134.10. No point is beyond these two values. All points are with in the limits. Hence this process is in control as per x bar chart.

Similalry, plotting the R bar chart. With batch number of x axis and the R bar value on the y axis.

The UCL and LCL for R bar chart are 31.87 and 3.23. So no point is lying beyond these two values. Hence the process is in control as per R bar chart as well.