APPENDIX
A: Comparison of Simulation-based and Numerical NPOC measures for 𝑁𝐿 = 3,000
B: Comparison of 𝑁𝑃𝑂𝐶(𝑡, 𝜋) and ∑𝑖,𝑗∈𝐼:𝑖<𝑗 𝑆𝑃𝑂𝐶(𝑡, 𝑙𝑖 , 𝑙𝑗 ) for Method 2 with three DFs
C: Comparison of 𝑁𝑃𝑂𝐶(𝑡, 𝜋) and ∑𝑖,𝑗∈𝐼:𝑖<𝑗 𝑆𝑃𝑂𝐶(𝑡, 𝑙𝑖 , 𝑙𝑗 ) for Method 2 with four DFs
D: Comparison of 𝑁𝑃𝑂𝐶(𝑡, 𝜋) and ∑𝑖,𝑗∈𝐼:𝑖<𝑗 𝑆𝑃𝑂𝐶(𝑡, 𝑙𝑖 , 𝑙𝑗 ) for Method 2 with five DFs
E: Comparison of 𝑁𝑃𝑂𝐶(𝑡, 𝜋) and ∑𝑖,𝑗∈𝐼:𝑖<𝑗 𝑆𝑃𝑂𝐶(𝑡, 𝑙𝑖 , 𝑙𝑗 ) for Method 3 with three DFs
F: Comparison of 𝑁𝑃𝑂𝐶(𝑡, 𝜋) and ∑𝑖,𝑗∈𝐼:𝑖<𝑗 𝑆𝑃𝑂𝐶(𝑡, 𝑙𝑖 , 𝑙𝑗 ) for Method 3 with four DFs
G: Comparison of 𝑁𝑃𝑂𝐶(𝑡, 𝜋) and ∑𝑖,𝑗∈𝐼:𝑖<𝑗 𝑆𝑃𝑂𝐶(𝑡, 𝑙𝑖 , 𝑙𝑗 ) for Method 3 with five DFs
H: Modified label-setting algorithm (LSA)
STEP0.
STEP1.
(i)
(ii)
(iii)
(iv)
Set 𝐿0 = {(0,0)} and 𝐿𝑖 = ∅ for all 𝑖 ∈ 𝑁\{0}
Initialize 𝐸𝑖 accordingly for each 𝑖 ∈ 𝑁
Set 𝑇𝑖 = ∅ for each 𝑖 ∈ 𝑁
for 𝑖 = 0,1, … , 𝑚
while 𝐸𝑖 \𝑇𝑖 ≠ ∅
choose 𝑘 ∈ 𝐸𝑖 \𝑇𝑖 so that 𝑊𝑖𝑘 is minimal
for all (𝑖, 𝑗) ∈ 𝐴 with 𝑊𝑖𝑘 + 𝑤𝑖𝑗 ≤ 𝑛
if (𝑊𝑖𝑘 + 𝑤𝑖𝑗 , 𝐶𝑖𝑘 + 𝑐𝑖𝑗 ) is not dominated
by (𝑊𝑗𝑙 , 𝐶𝑗𝑙 ) for any 𝑙 ∈ 𝐸𝑗
then set 𝐿𝑗 = 𝐿𝑗 ∪ {(𝑊𝑖𝑘 + 𝑤𝑖𝑗 , 𝐶𝑖𝑘 + 𝑐𝑖𝑗 )}
if 𝑾𝒌𝒊 + 𝒘𝒊𝒋 = 𝒏 and 𝑪𝒌𝒊 + 𝒄𝒊𝒋 < 𝑈 (upper bound)
then 𝑼 = 𝑪𝒌𝒊 + 𝒄𝒊𝒋
update 𝐸𝑗 accordingly
set 𝑇𝑖 = 𝑇𝑖 ∪ {𝑘}.
The algorithm starts with initial label (0,0) associated with start node 0 and extends the set
of all labels by treating an existing label associated with a selected node. For each node 𝑖 ∈
𝑀\{𝑚 + 1}, the algorithm treats all labels associated with that node (STEP1 (i)), selecting a
label with minimum 𝑊𝑖𝑘 . The treatment of label (𝑊𝑖𝑘 , 𝐶𝑖𝑘 ) considers each outgoing arc (𝑖, 𝑗)
that does not exceed the number of DFs (i.e., 𝑊𝑖𝑘 + 𝑤𝑖𝑗 ≤ 𝑛) (STEP1 (ii)). A new label
(𝑊𝑖𝑘 + 𝑤𝑖𝑗 , 𝐶𝑖𝑘 + 𝑐𝑖𝑗 ) associated with node 𝑗 is added only if it is not dominated by any other
labels on node 𝑗 (STEP1 (iii)). If the added label is weight feasible and the cumulative cost is
less than the current upper bound, the upper bound is improved (STEP 1 (iv)).
I. Results for cases with 1 × 1𝑘𝑚2 and . 5 × .5𝑘𝑚2 cells
(1)
(2)
(3)
(4)
(5)
METHOD 1
case
1
2
3
4
5
6
7
8
9
10
11
12
(6)
(7)
(8)
(9)
METHOD 2
(10)
(11)
(12)
METHOD 3
DFs
𝑍1∗
RT (secs.)
𝑍2∗
RT (secs.)
TR

𝑍3∗
RT (secs.)
TR

3
3.46
1.04
3.37
0.06
16.79
2.62
3.37
0.03
34.70
2.62
4
4.19
1.78
4.19
0.18
9.88
0.00
4.19
0.03
59.27
0.00
5
4.78
1.29
4.78
0.21
6.14
0.02
4.65
0.03
43.00
2.64
3
4.41
2.89
4.36
0.15
19.25
1.22
4.36
0.02
144.35
1.22
4
5.62
8.29
5.59
0.84
9.87
0.45
5.42
0.02
414.60
3.59
5
6.49
10.71
6.47
1.78
6.02
0.21
6.19
0.02
535.70
4.63
3
3.82
1.69
3.74
0.08
21.06
1.91
3.74
0.02
84.25
1.91
4
4.59
3.49
4.55
0.35
10.06
0.89
4.54
0.02
174.55
0.90
5
5.28
2.70
5.25
0.45
6.00
0.55
5.22
0.02
135.00
1.22
3
2.83
3.11
2.81
0.15
20.73
0.71
2.81
0.03
103.67
0.71
4
3.65
9.38
3.65
0.95
9.89
0.00
3.45
0.03
312.67
5.24
5
4.22
12.29
4.13
2.07
5.94
2.06
4.06
0.03
409.77
3.86
3
2.48
2.76
2.42
0.12
22.65
2.39
2.38
0.02
138.15
3.85
4
3.24
8.26
3.21
0.81
10.18
1.16
3.09
0.02
412.95
4.77
5
3.78
10.37
3.76
1.73
5.99
0.57
3.57
0.02
518.65
5.48
3
3.22
4.09
3.14
0.20
20.14
2.43
3.12
0.03
136.30
3.12
4
4.17
12.44
4.17
1.25
9.95
0.02
4.11
0.03
414.63
1.58
5
4.92
17.17
4.90
2.87
5.98
0.47
4.73
0.03
572.30
3.87
3
14.12
562.04
13.98
28.75
19.55
0.99
13.98
0.56
1001.85
0.99
4
20.83
5105.87
19.47
523.56
9.75
6.53
19.20
0.55
9368.56
7.84
5
21.07
18673.43
20.83
3175.57
5.88
1.13
20.83
0.56
33285.97
1.10
3
17.97
574.19
17.65
29.13
19.71
1.79
17.58
0.56
1021.70
2.17
4
23.63
5480.94
23.63
554.90
9.88
0.00
22.22
0.55
10038.35
5.97
5
27.30
20303.37
27.25
3410.87
5.95
0.16
27.25
0.55
37185.66
0.16
3
16.74
589.59
16.53
30.37
19.41
1.26
16.53
0.56
1049.09
1.26
4
21.10
5347.78
20.88
546.86
9.78
1.02
20.40
0.56
9532.58
3.29
5
24.72
19346.19
24.30
3281.42
5.90
1.70
23.62
0.58
33528.93
4.44
3
11.54
775.00
11.40
39.42
19.66
1.19
11.40
0.66
1183.20
1.19
4
14.93
8227.46
14.68
840.57
9.79
1.64
14.50
0.66
12541.86
2.84
5
17.49
33938.44
16.98
5760.21
5.89
2.88
16.22
0.66
51814.42
7.24
3
10.20
532.09
9.91
27.00
19.70
2.93
9.96
0.50
1066.31
2.36
4
13.96
5237.68
13.89
532.04
9.84
0.45
13.08
0.51
10190.04
6.24
5
16.40
20491.54
16.08
3462.80
5.92
1.91
15.77
0.51
39866.81
3.83
3
14.05
614.18
13.52
31.53
19.48
3.77
12.94
0.59
1035.72
7.88
4
17.90
5680.74
17.88
582.62
9.75
0.15
17.79
0.59
9579.65
0.65
5
21.31
20344.32
20.84
3464.07
5.87
2.20
20.44
0.58
35258.78
4.08
Column 1 gives the case number. For each case, column 2 provides 𝑛, the number of DFs
to be deployed. Columns 3-4, 5-8, and 9-12 detail results associated with methods 1, 2, and 3,
respectively, giving, for each method, the objective function value (𝑍 ∗ ), which is transformed
into the corresponding numerical POC value, and run time (RT) in seconds. In addition, for
Methods 2 and 3, columns 7 and 11 give the run time ratio (TR) relative to Method 1 and
columns 8 and 12 provide relative deviation ∆ (in %) of objective function values in
comparison with Method 1 (column 3). For Method 3, column 10 includes times to compute
arc costs, build the model network, and run the modified LSA.
J. Comparisons of objective function values for . 5 × .5km2 cells
K. Comparisons of run times for . 5 × .5km2 cells
(a) Run times
L: Deployments of three DFs obtained from Method 3 (black dots represent DF locations selected)
#1: convex, left-skewed, 1 × 1km2
#2: convex, centered, 1 × 1km2
#3: convex, right-skewed, 1 × 1km2
#4: concave, left-skewed, 1 × 1km2
#5: concave, centered, 1 × 1km2
#6: concave, right-skewed, 1 × 1km2
#7: convex, left-skewed, . 5 × .5km2
#9: convex, right-skewed, . 5 × .5km2
#11: concave, centered, . 5 × .5km2
#8: convex, centered, . 5 × .5km2
#10: concave, left-skewed, . 5 × .5km2
#12: concave, right-skewed, . 5 × .5km2
#13: convex, left-skewed, . 25 × .25km2
#14: convex, centered, . 25 × .25km2
#15: convex, right-skewed, . 25 × .25km2
#16: concave, left-skewed, . 25 × .25km2
#17: concave, centered, . 25 × .25km2
#18: concave, right-skewed, . 25 × .25km2