16.4 Analyze Phase
The Analyze phase has five steps:

Develop a more detailed process map (that is, more detailed than the process map developed in the SIPOC analysis of the Define phase).

Construct operational definitions for each input or process variable (called Xs).

Perform a Gage R&R study on each X (test the adequacy of the measurement system).

Develop a baseline for each X.

Develop hypotheses between the Xs and Ys.
The Ys are the output measures used to determine whether the CTQs are met.
Team members prepare a detailed process map identifying and linking the Xs and Ys, as shown in Figure 16.11.
Figure 16.11 Process Map Linking CTQs and Xs for the MSD Purchasing Process
Team members develop an operational definition for each X variable identified on the process map. The operational definitions for X_{1}, X_{2}, X_{3}, and X_{8} relate to individual MSDs and are shown below.

Criteria: Each X conforms to either one or the other of the options.
X_{1}
Vendor
Ibix
Office Optimum
X_{2}
Size
Small (stock size)
Large (stock size)
X_{3}
Ridges
With ridges
Without ridges
X_{8}
Type of usage
Large stack of paper (number of papers is 10 or more)
Small stack of paper (number of papers is 9 or less)

Test: Select MSD.

Decision: Determine X_{1}, X_{2}, X_{3}, and X_{8} options for the selected MSD.

The operational definitions for the procedures used to measure X_{4}, X_{5}, X_{6}, and X_{7} are shown below.

Criteria: Compute the cycle time in days by subtracting the order date from the date on the bill of lading.
X_{4}
Cycle time from order to receipt for MSDs
In days

Test: Select a box of MSDs upon receipt of shipment from vendor. Compute the cycle time.

Decision: Determine X_{4} for the selected box of MSDs.

Criteria: Count the number of boxes of MSD received for a given order. Subtract the number of boxes ordered from the number of boxes received for the order under study.
X_{5}
Discrepancy in count from order placed and order received
In boxes of MSDs by order

Test: Select a particular purchase order for MSDs.

Decision: Compute the value of X_{5} in number of boxes for the selected purchase order.

Criteria: Compute the cycle time in days to place a shipment of MSDs in inventory by subtracting the date the shipment was received from the date the order was placed in inventory.
X_{6}
Cycle time to place product in inventory
In days

Test: Select a particular purchase order.

Decision: Compute the value of X_{6} in days for the selected purchase order.

Criteria: Compute the inventory shelftime in days for a box of MSDs by subtracting the date the box was placed in inventory from the date the box was removed from inventory.
X_{7}
Inventory shelf time
In days

Test: Select a box of MSDs.

Decision: Compute the value of X_{7} in days for the selected box of MSDs.
Team members conduct Gage R&R studies for the Xs. Recall that the purpose of a Gage R&R study is to determine the adequacy of the measurement system for an X. In this case, the measurement systems for all of the Xs are known to be reliable and reproducible. Hence, Gage R&R studies were not conducted by team members.
Team members gather baseline data on durability (Y_{1}) functionality (Y_{2}), and the relevant Xs using the following sampling plan. For a 2week period, the first box of MSDs brought to the PSD each hour was selected as a sample. This yielded a sample of 80 boxes of MSDs, which can be seen Table 16.20.
Table 16.20. Baseline Data
Sample 
Day 
Hour 
X_{1} 
X_{2} 
X_{3} 
X_{7} 
Dur 
Fun 

1 
Mon 
1 
1 
0 
0 
7 
2 
5 
2 
Mon 
2 
0 
1 
0 
7 
2 
9 
3 
Mon 
3 
0 
0 
1 
7 
10 
7 
4 
Mon 
4 
0 
1 
0 
7 
1 
4 
5 
Mon 
5 
0 
0 
0 
7 
7 
3 
6 
Mon 
6 
0 
1 
1 
7 
2 
5 
7 
Mon 
7 
0 
1 
1 
7 
1 
9 
8 
Mon 
8 
0 
0 
0 
7 
7 
5 
9 
Tue 
1 
0 
1 
0 
8 
2 
8 
10 
Tue 
2 
0 
1 
0 
8 
1 
7 
11 
Tue 
3 
0 
1 
0 
8 
1 
13 
12 
Tue 
4 
1 
1 
1 
8 
9 
5 
13 
Tue 
5 
1 
1 
0 
8 
9 
9 
14 
Tue 
6 
1 
1 
1 
8 
10 
11 
15 
Tue 
7 
1 
1 
1 
8 
10 
11 
16 
Tue 
8 
0 
0 
1 
8 
8 
9 
17 
Wed 
1 
1 
1 
1 
9 
8 
11 
18 
Wed 
2 
1 
0 
0 
9 
1 
11 
19 
Wed 
3 
1 
1 
1 
9 
10 
11 
20 
Wed 
4 
0 
0 
0 
9 
7 
11 
21 
Wed 
5 
1 
1 
1 
9 
9 
9 
22 
Wed 
6 
0 
0 
1 
9 
9 
5 
23 
Wed 
7 
1 
0 
1 
9 
2 
11 
24 
Wed 
8 
1 
0 
0 
9 
1 
10 
25 
Thu 
1 
1 
0 
1 
10 
1 
14 
26 
Thu 
2 
0 
1 
1 
10 
1 
10 
27 
Thu 
3 
1 
1 
1 
10 
8 
13 
28 
Thu 
4 
0 
0 
1 
10 
10 
12 
29 
Thu 
5 
0 
0 
0 
10 
7 
14 
30 
Thu 
6 
0 
1 
1 
10 
3 
13 
31 
Thu 
7 
0 
0 
0 
10 
9 
13 
32 
Thu 
8 
1 
1 
1 
10 
8 
11 
33 
Fri 
1 
0 
1 
0 
1 
2 
0 
34 
Fri 
2 
0 
1 
0 
1 
2 
1 
35 
Fri 
3 
0 
1 
0 
1 
1 
6 
36 
Fri 
4 
0 
1 
0 
1 
3 
3 
37 
Fri 
5 
0 
1 
0 
1 
2 
2 
38 
Fri 
6 
1 
1 
0 
1 
10 
6 
39 
Fri 
7 
0 
0 
1 
1 
10 
0 
40 
Fri 
8 
0 
1 
0 
1 
2 
0 
41 
Mon 
1 
0 
1 
1 
4 
3 
4 
42 
Mon 
2 
0 
1 
0 
4 
3 
7 
43 
Mon 
3 
0 
1 
1 
4 
3 
3 
44 
Mon 
4 
0 
0 
0 
4 
10 
2 
45 
Mon 
5 
1 
1 
0 
4 
8 
5 
46 
Mon 
6 
0 
1 
1 
4 
3 
4 
47 
Mon 
7 
1 
0 
0 
4 
1 
4 
48 
Mon 
8 
0 
0 
1 
4 
10 
5 
49 
Tue 
1 
1 
1 
1 
5 
11 
6 
50 
Tue 
2 
1 
0 
1 
5 
3 
4 
51 
Tue 
3 
1 
1 
0 
5 
10 
6 
52 
Tue 
4 
1 
0 
1 
5 
3 
5 
53 
Tue 
5 
1 
0 
0 
5 
2 
4 
54 
Tue 
6 
0 
0 
0 
5 
9 
5 
55 
Tue 
7 
0 
0 
1 
5 
9 
5 
56 
Tue 
8 
0 
1 
0 
5 
3 
7 
57 
Wed 
1 
0 
0 
1 
6 
9 
5 
58 
Wed 
2 
1 
1 
0 
6 
9 
7 
59 
Wed 
3 
0 
0 
0 
6 
9 
5 
60 
Wed 
4 
1 
0 
0 
6 
2 
6 
61 
Wed 
5 
1 
0 
1 
6 
2 
5 
62 
Wed 
6 
1 
1 
1 
6 
10 
5 
63 
Wed 
7 
0 
1 
0 
6 
1 
7 
64 
Wed 
8 
0 
1 
0 
6 
2 
5 
65 
Thu 
1 
0 
0 
1 
7 
10 
7 
66 
Thu 
2 
1 
1 
0 
7 
9 
5 
67 
Thu 
3 
1 
0 
0 
7 
1 
7 
68 
Thu 
4 
0 
1 
0 
7 
2 
5 
69 
Thu 
5 
1 
0 
1 
7 
1 
6 
70 
Thu 
6 
0 
1 
0 
7 
1 
5 
71 
Thu 
7 
1 
0 
0 
7 
1 
8 
72 
Thu 
8 
1 
1 
1 
7 
10 
5 
73 
Fri 
1 
0 
1 
1 
8 
3 
7 
74 
Fri 
2 
1 
1 
1 
8 
9 
7 
75 
Fri 
3 
1 
0 
0 
8 
1 
13 
76 
Fri 
4 
0 
1 
1 
8 
2 
8 
77 
Fri 
5 
0 
1 
1 
8 
3 
9 
78 
Fri 
6 
1 
1 
1 
8 
8 
10 
79 
Fri 
7 
1 
0 
1 
8 
3 
11 
80 
Fri 
8 
0 
0 
1 
8 
10 
11 
X_{1} = vendor (0 = Office Optimum and 1 = Ibix) X_{2} = size (0 = small and 1 = large) X_{3} = ridges (0 = without and 1 = with) X_{7} = inventory shelftime, in days 
For each sampled box, team members determined the durability (Y_{1}) and functionality (Y_{2}) measurements. Furthermore, information concerning the vendor (X_{1}), size of the MSD (X_{2}), whether the MSD has ridges (X_{3}), and inventory shelflife is recorded (X_{7}).
The Purchasing Department will separately study cycle time from order to receipt of order (X_{4}), discrepancy between ordered and received box counts (X_{5}), and cycle time from receipt of order to placement in inventory (X_{6}). These last factors may influence such concerns as choice of vendor, ordering procedures, and inventory control, but they do not impact durability and functionality. Furthermore, the MSDs are not tested after they are used, so the type of usage (X_{8}) is not studied here. As was indicated in the Define phase, certain variables (e.g., X_{4}, X_{5}, X_{6}, and X_{7}) can be addressed in subsequent Six Sigma projects.
The baseline data revealed that the yield for durability is 0.4625 (37/80) and the yield for functionality is 0.425 (34/80), as shown in Table 16.21. As before, this indicates very poor levels for the CTQs in the PSD. For comparison purposes, the judgment sample carried out by the team during the Define phase showed that the yield was approximately 40% (i.e., the team assumed the failure rate was approximately 60%) for both durability and functionality. The slightly increased yields in this study can be due to natural variation in the process. The baseline data also showed that 56.25% of all MSDs are from Office Optimum (X_{1}), 42.50% of MSDs are small (X_{2}), 50.00% of all MSDs are without ridges (X_{3}), and the average shelftime for boxes of MSDs (X_{7}) is 6.5 days, with a standard deviation of 2.5 days (see Table 16.21).
Table 16.21. Basic Statistics on Baseline Data
Variable 
Proportion 
Mean 
Standard deviation 


Y_{1}: Durability 
Four or more bends/clip 
0.4625 
5.213 
3.703 
Y_{2}: Functionality 
Five or fewer broken/box 
0.4250 
7.025 
3.438 
X_{1}: Vendor 
0 = Office Optimum 
0.5625 

1 = Ibix 
0.4375 

X_{2}: Size 
0 = Small 
0.4250 

1 = Large 
0.5750 

X_{3}: Ridges 
0 = Without ridges 
0.5000 

1 = With ridges 
0.5000 

X_{7}: Inventory shelftime 
Shelftime in days 
6.5000 
2.5160 
Team members develop hypotheses [Y = f(X)] about the relationships between the Xs and the Ys to identify the Xs that are critical to improving the center, spread, and shape of the Ys with respect to customer specifications. This is accomplished through data mining. Data mining is a method used to analyze passive data; that is, data that is collected as a consequence of operating a process. In this case, the baseline data in Table 16.20 is the passive data set that will be subject to data mining procedures. Dot plots or box plots of durability (Y_{1}) and functionality (Y_{2}) stratified by X_{1}, X_{2}, X_{3}, and X_{7} can be used to generate some hypotheses about main effects (i.e., the individual effects of each X on Y_{1} and Y_{2}). Interaction plots can be used to generate hypotheses about interaction effects (i.e., those effects on Y_{1} or Y_{2} for which the influence of one X variable depends on the level or value of another X variable) if all combinations of levels of X variables are studied. If not all combinations of levels of X variables are studied, then interaction effects are often not discovered.
Team members constructed dot plots from the baseline data in Table 16.20 to check whether any of the Xs (i.e., main effects) impact durability (Y_{1}) and functionality (Y_{2}). The dot plots for durability are shown in Figures 16.12–16.15. The dot plots for functionality are shown in Figures 16.16–16.19.
Figure 16.12 Minitab Dot Plot for Durability by X1 (i.e., Vendor)
Figure 16.15 Minitab Dot Plot for Durability by X7 (i.e., Shelflife)
Figure 16.13 Minitab Dot Plot for Durability by X2 (i.e., Size)
Figure 16.14 Minitab Dot Plot for Durability by X3 (i.e., Ridges)
Figure 16.16 Minitab Dot Plot for Functionality by X1 (i.e., Vendor)
Figure 16.19 Minitab Dot Plot for Functionality by X7 (i.e., Shelflife)
Figure 16.17 Minitab Dot Plot for Functionality by X2 (i.e., Size)
Figure 16.18 Minitab Dot Plot for Functionality by X3 (i.e., Ridges)
The dot plots for durability (Y_{1}) indicate: (1) the values of durability tend to be low or high, with a significant gap between 4 and 6 for X_{1}, X_{2}, X_{3}, and X_{7}, and (2) the variation in durability is about the same for all levels of X_{1}, X_{2}, X_{3}, and X_{7}. The dot plots for functionality (Y_{2}) indicate: (1) the values of functionality tend to be lower when X_{1} = 0 than when X_{1} = 1, (2) the variation in functionality is about the same for all levels of X_{2} and X_{3}, and (3) the values of functionality tend to be lower for low values of X_{7}.
Discussion of the Analysis of Durability
Because there are no clear differences in variation (i.e., spread) of durability for each of the levels of X_{1}, X_{2}, X_{3}, and X_{7}, the team wondered whether there might be differences in the average (i.e., center) for each level of the individual Xs. Team members constructed a main effects plot for durability to study differences in averages (see Figure 16.20).
Figure 16.20 Minitab Main Effects Plot for Durability by X1, X2, X3, and X7.
Figure 16.20 indicates that for the ranges of shelflife observed, there is no clear pattern for the relationship of shelflife (X_{7}) to the average durability. On the other hand, it appears that ridges (i.e., X_{3} = 1) have a positive relationship to the average durability. At first glance, it would seem that better results for average durability are seen when the vendor Ibix is chosen using small MSDs (i.e., X_{1} = 1 and X_{2} = 0), whereas using large MSDs from Office Optimum (i.e., X_{1} = 0 and X_{2} = 1) yields worse results.
While discussing the dot plots and main effects plot, it is dangerous to make any conclusions without knowing whether there are interaction effects. An interaction effect is present when the amount of change introduced by changing one of the Xs depends on the value of another X. In that case, it is misleading to choose the best value of the Xs individually without first considering the interactions between the Xs. Consequently, team members did an interaction plot for X_{1}, X_{2}, and X_{3}. X_{7} was not included in the interaction plot because the main effects plot indicated no clear pattern or relationship with durability (Y_{1}). All combinations of levels of the X variables must be present to draw an interaction plot. This is often not the case with passive data (i.e., no plan was put in place to insure all combinations were observed in the datagathering phase). Fortunately, although not all combinations were observed equally often, they were all present. Figure 16.21 is the interaction plot for durability.
Figure 16.21 Minitab Interaction Effects Plot for Durability by X1, X2, and X3
Surprise! The interaction plot indicates that there is a possible interaction between X_{1} (i.e., vendor) and X_{2} (i.e., size). How is this known? When there is no interaction, the lines should be parallel to each other, indicating that the amount of change in average durability when moving from one level of each X variable to another level should be the same for all values of another X variable. This plot shows the lines on the graph of X_{1} and X_{2} not only are not parallel, but they cross. The average durability is the highest when either large Ibix MSDs (i.e., X_{1} = 1 and X_{2} = 1) or small Office Optimum MSDs (i.e., X_{1} = 0 and X_{2} = 0) are used. This means the choice of vendor may depend on the size of MSD required. The main effects plot suggests that the best results for average durability occurs when small MSDs from Ibix are used, but the interactions plot suggests this combination yields a bad average durability. To study all of this further, the team decides that during the Improve phase, they will run a full factorial design to examine the relationship of X_{1}, X_{2}, and X_{3} on durability (Y_{1}) because the main effects plot indicates potential patterns. Again, there does not appear to be a relationship between durability (Y_{1}) and X_{7}.
Discussion of the Analysis of Functionality
Figures 16.22 and 16.23 show the main effects and interaction effects plots for functionality (Y_{2}).
Figure 16.22 Minitab Main Effects Plot for Functionality by X1, X2, X3, and X7.
Figure 16.23 Minitab Interaction Effects Plot for Functionality by X1, X2, and X3
The main effects plot indicates that higher values of shelflife (X_{7}) yield higher values for functionality (Y_{2}). The team surmised that the longer a box of MSDs sets in inventory (i.e., higher values of shelflife), the higher will be the count of broken MSDs (i.e., functionality will be high). From a practical standpoint, the team felt comfortable with this conclusion. They decided the Purchasing Department should put a Six Sigma project in place to investigate whether the potential benefit of either a "justintime" MSD ordering process or the establishment of better inventory handling procedures will solve the problem.
The interaction effect plot indicates a potential interaction between the X_{2} (i.e., size) and X_{3} (i.e., ridges). Better results for functionality (i.e., low values) were observed for large MSDs without ridges (i.e., X_{2} = 1 and X_{3} = 0). Why this may be the case needs to be studied further. Also, there may be an interaction between X_{1} (i.e., vendor) and X_{2} (i.e., size), but it appears that better results are observed whenever Office Optimum is used (i.e., X_{1} = 0). In other words, the average count of broken MSDs is lower (i.e., functionality average is lower) whenever Office Optimum is the vendor.
Analyze Phase Summary
The Analyze phase resulted in the following hypotheses:

Hypothesis 1: Durability = f(X_{1} = Vendor, X_{2} = Size, X_{3} = Ridges) with a strong interaction effect between X_{1} and X_{2}.

Hypothesis 2: Functionality = f(X_{1} = vendor, X_{2} = size, X_{3} = ridges, X_{7} = shelflife), the primary driver being X_{7} with some main effect due to X_{1} and an interaction effect between X_{2} and X_{3}.
X_{7} is the main driver of the distribution of functionality (Y_{2}) and is under the control of the employees of POI. Hence, team members restructured Hypothesis 2 as follows: Functionality = f(X_{1} = vendor, X_{2} = size, X_{3} = ridges) for each fixed level of X_{7} (shelflife).