CHAPTER 4 SEARCH RANGE (SR) ASSIGNMENT
4.4 Simulation Experiment
4.4.3. Phase II Experiment
Two factors are designed to evaluate the appropriate SR under a different number of IV (v) and WF (f), and only the WFs or IVs in the assigned range will be considered when dispatching occurs.
The dispatching procedure is shown in Figure 4. 3(b). The factors and levels selected were as follows.
(a) Factor A: Moving Rate (MR); numeric factor, three levels. Levels of MR were MRi, i = 1, 2, 3, where MR1, MR2, MR3is 70 lots/hr, 105 lots/hr, 140 lots/hr respectively.
(b) Factor B: Search Range (SR); categorical factor, six levels. Levels of SR were SRj, j = 1, 1.5, 2, 2.5, 3, 0. The SR definition was made as follows, and the levels of SR are shown in Table 4. 4
(1) FSRi,j,v: the SR of FSV under MRiand SRjwhen IV number is v.
Table 4. 4 Levels of SR assignment
FSRi,j,v(m)
i=1 i=2 i=3
j j j
v
0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 0
1 111 131 152 172 193 213 608.5 111 132 152 173 194 214 608.5 114 135 156 177 198 219 608.5 2 86 104 121 138 155 173 608.5 86 104 121 139 156 174 608.5 87 105 123 142 160 178 608.5 3 74 90 105 121 136 152 608.5 74 90 105 121 137 153 608.5 74 90 106 122 138 154 608.5 4 67 82 96 111 125 139 608.5 67 81 96 111 126 140 608.5 65 80 95 109 124 138 608.5 5 61 75 88 102 115 129 608.5 61 74 88 101 115 129 608.5 61 74 88 102 115 129 608.5 6 57 70 83 95 108 121 608.5 56 69 82 95 107 120 608.5 56 69 82 95 108 120 608.5 7 54 66 78 90 102 114 608.5 53 66 78 90 102 114 608.5 52 65 77 89 101 113 608.5 8 52 63 75 87 98 110 608.5 51 62 74 86 97 109 608.5 49 61 73 84 96 108 608.5 9 49 60 71 83 94 105 608.5 47 58 69 80 92 103 608.5 47 58 69 80 91 102 608.5
VSRi,j,f(m)
i=1 i=2 i=3
j j j
f
0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 0
1 123 144 165 186 207 228 608.5 122 145 167 189 212 234 608.5 124 146 169 192 215 238 608.5 2 99 120 141 163 184 205 608.5 100 121 143 164 185 206 608.5 100 122 144 165 187 209 608.5 3 81 99 117 136 154 172 608.5 87 108 128 148 168 188 608.5 87 108 128 149 169 190 608.5 4 76 94 113 131 149 167 608.5 81 101 121 141 161 181 608.5 79 99 118 138 157 176 608.5 5 65 81 97 113 130 146 608.5 73 92 112 131 151 170 608.5 74 93 112 132 151 170 608.5 6 36 46 56 66 76 86 608.5 68 86 105 123 141 159 608.5 70 88 106 125 143 161 608.5
7 73 91 110 129 147 166 608.5 64 82 99 117 134 151 608.5
8 50 64 78 92 106 119 608.5 60 77 94 111 128 145 608.5
9 55 71 87 104 120 136 608.5 57 74 91 107 124 141 608.5
10 42 54 65 77 89 101 608.5 55 71 87 103 120 136 608.5
11 30 32 33 34 36 37 608.5 52 69 85 101 117 133 608.5
12 49 64 79 94 109 124 608.5
13 47 62 78 94 109 125 608.5
14 55 71 87 103 120 136 608.5
15 49 64 79 95 110 126 608.5
16 55 70 86 101 117 132 608.5
17 29 40 51 62 73 84 608.5
18 29 38 46 55 63 71 608.5
A traditional statistical experimental design with two-factor full-factorial (Montgomery [45]) was used. The number of scenarios was 3(MR)*6(SR) = 18, and the number of experiments performed was 18(combination)*10(replications) = 180. The simulation interval was 30 days, and warm-up was 2 days.
B. Experiment Result and Discussion
The ANOVA analysis as summarised in Table 4.5 indicates that the MR significantly affects all indices. Also, the SR and the interaction significantly affect all indices except the TP at 95%
confidence level. The response trend under levels of MR and SR can be observed from the interaction graphs, as depicted in Figure 4.8.
Table 4.5 P-values in SR phase II experiment
Factors Indices
MR SR MR2 MR*SR R2
TP (lots) < 0.0001* 0.4611 0.0122* 0.2997 0.9999
WT (sec) < 0.0001* < 0.0001* < 0.0001* < 0.0001* 0.9715 stdDT (sec) < 0.0001* < 0.0001* < 0.0001* < 0.0001* 0.9825 Vemp (%) < 0.0001* < 0.0001* < 0.0001* < 0.0001* 0.9996
*=significant at 95% confidence level
35
25
15
5 100
85
70 55 40
130 105
80 55 30
110 92.5
75 57.5 40
SR1SR1.5SR2SR2.5SR3 SR0 B: SR
SR1SR1.5SR2SR2.5SR3 SR0 B: SR
SR1SR1.5SR2SR2.5SR3 SR0 B: SR
SR1SR1.5SR2SR2.5SR3 SR0 B: SR
(a) TP (lots) (unit: 1000) (b) WT (sec)
(c) stdDT (sec) (d) Vemp (%) A: MR1
A: MR2 A: MR3
A: MR1 A: MR2
A: MR3
A: MR1 A: MR2 A: MR3
A: MR1 A: MR2 A: MR3
Figure 4.8 The Interaction graphs of MR and SR
Further, the Post Hoc Multiple Comparisons was done with the least significant difference (LSD) method to compare all pairs of the six SR under each of the three MR, as summarised in Table 4.6. The value is the mean of performance, measured from the 10 replications, and the rank in the different alphabet means the effects of SR were significant at 95% confidence level. Table 4.6 implied that the shorter SR made for a longer WT because the narrow range decreased the chance for the vehicle to find the WFs, and then the WFs need more time to wait for being assigned; the
Vemp would be lower (better) and indicates vehicle work more efficiently while SR is shorter.
However, no SR outperforms the others in all indices.
Table 4.6 Post Hoc Multiple Comparisons in SR phase II experiment Indices
TP (lots) WT (sec) stdDT (sec) Vemp (%)
MR
(lots/hr) SR
rank mean rank mean rank mean rank mean
SR1 A 46016.6 F 126.461 E 110.121 A 10.141
SR1.5 A 46016.5 E 69.673 D 60.955 C 10.296
SR2 A 46016.8 D 50.376 C 47.398 BC 10.251
SR2.5 A 46016.7 C 42.346 B 43.070 AB 10.216
SR3 A 46016.4 B 36.860 A 41.220 AB 10.196
70
SR0 A 45965.8 A 34.192 A 40.803 AB 10.203
SR1 A 68946.5 D 76.295 D 81.238 A 17.167
SR1.5 A 69011.8 C 49.328 C 60.999 B 17.951
SR2 A 68946.9 B 43.230 B 55.840 C 18.116
SR2.5 A 68947.1 A 42.111 A 53.778 D 18.195
SR3 A 68946.7 A 41.930 A 53.278 D 18.182
105
SR0 A 68948.8 A 41.565 A 53.178 CD 18.173
SR1 A 91687.7 C 82.731 C 100.732 A 25.629
SR1.5 A 91890.4 B 69.161 B 89.929 B 27.789
SR2 A 91687.9 A 66.607 A 87.376 C 28.579
SR2.5 A 91688.5 A 65.974 A 86.772 D 28.849
SR3 A 91687.6 A 65.944 A 86.638 D 28.886
140
SR0 A 91824.9 A 65.611 A 86.744 D 28.866
Therefore, a multiple response method called desirability (Myers & Montgomery, Stat Ease) [46, 56] is used to integrate multiple indices into one. The method makes use of an objective function D(X), called the desirability function.
d d d d
D
in
i n
n n
1 1 1 2
1
where n is the number of responses; di is the desirable range for each response, 0≦di≦1; and D(X) is a geometric mean of all transformed responses. The desirability of SRj under MRi is shown in Figure 4.9, and some discussions are as follows.
SR1 SR1.5SR2 SR2.5SR3 SR0 B: SR
SR1 SR1.5SR2 SR2.5SR3 SR0 B: SR
SR1 SR1.5SR2 SR2.5SR3 SR0 B: SR
0.24 0.52
0.64
0.69 0.69
Prediction 0.73
0.20
0.51 0.47 0.48 0.49 Prediction 0.55
0.34
0.45 0.39 0.38 0.42 Prediction 0.57
0.000.250.500.751.00Desirability
(a) A: MR1(70 lots/hr) (b) A: MR2(105 lots/hr) (c) A: MR3(140 lots/hr)
Figure 4.9 The desirability of SR for each MR
(1) SR1 is not applicable in any system loading. Too narrow a range decreases the chance for dispatching successfully, and then makes more WT and higher stdDT for FOUP, even if it can enable lower Vemp; see Figure 4.9 and Table 4.6.
(2) In a light system (fewer WFs) such as MR1, a shorter SR is not appropriate. Responses to the phase I result in fewer WFs making a longer DVemp. Hence a longer SR such as SR2.5or SR3or SR0makes for better performance; see Figure 4.9(a).
(3) In a heavy system (more WFs) such as MR2and MR3, the longer SR is not required. Responses to the phase I result in more WFs making a shorter DVemp. Hence a shorter SR such as SR1.5
makes for better performance; see Figure 4.9(b), (c).
(4) The WFs or IVs far from V or F could be ignored by setting an appropriate SR. For instance, ignoring 7.5% (4.8% + 2.1% + 0.6%) IVs for FSV and 7.5% (5.3% + 1.9% + 0.3%) WFs for VSF, of which the DVemp is longer than the interval +1.5σunder MR3to improve performance;
see Figure 4.7(e), (f) and Figure 4.9(c).
(5) The result of testing if SR is required also can be seen. SR0might be used in a light system such as MR1 because of its simpler control logic and performance is close to the optimal SR3; see Figure 4.9(a). However, if system loading is increasing to MR2 or MR3, SR0 is not applicable;
see Figure 4.9(b), (c).
Finally, the model equations in terms of coded factors (because there is a category factor) can be used to predict the response at a interesting point, where code for factor A: MR is -1, 0, 1 at MR1, MR2, MR3 respectively, and B[1],B[2],…,B[5]isto representfactor B: SR as SR1, SR1.5,…,SR3
respectively. The code for factor B: SR is 0 or 1. The model equations also can provide guidance for practitioners in selecting the preferable setting based on the changeable environment and performance measures.
(1) TP (lot) =
68958.0 + 22868.2*A - 19.9*B[1] + 69.4*B[2] - 19.7*B[3] - 19.4*B[4] - 20.0*B[5] - 81.6*A2 -32.6*AB[1] + 68.8*AB[2] - 32.6*AB[3] - 32.3*AB[4] - 32.6*AB[5]
(2) WT (sec) =
49.1 + 4.7*A + 35.7*B[1] + 3.3*B[2] - 6.1*B[3] - 9.3*B[4] - 11.2*B[5] + 15.6*A2- 26.5*AB[1]
- 4.9*AB[2] + 3.4*AB[3] + 7.1*AB[4] + 9.9*AB[5]
(3) stdDT (sec) =
59.7 + 16.2*A + 28.5*B[1] + 1.7*B[2] - 5.4*B[3] - 7.7*B[4] - 8.5*B[5] + 13.8*A2- 20.9*AB[1]
- 1.7*AB[2] + 3.8*AB[3] + 5.6*AB[4] + 6.5*AB[5]
(4) Vemp (%) =
17.96% + 8.94%*A - 1.12%*B[1] - 0.08%*B[2] + 0.22%*B[3] + 0.33%*B[4] + 0.33%*B[5] + 1.19%*A2- 1.20%*AB[1] - 0.20%*AB[2] + 0.22%*AB[3] + 0.38%*AB[4] + 0.40%*AB[5]