![](bg4.png)
R Torque
10 450 109 110 111 112 113 114 115 116 117 118 119 120
9 400 97 98 99 100 101 102 103 104 105 106 107 108
8 350 85 86 87 88 89 90 91 92 93 94 95 96
7 300 73 74 75 76 77 78 79 80 81 82 83 84
6 250 61 62 63 64 65 66 67 68 69 70 71 72
5 200 49 50 51 52 53 54 55 56 57 58 59 60
4 150 37 38 39 40 41 42 43 44 45 46 47 48
3 100 25 26 27 28 29 30 31 32 33 34 35 36
2 50 13 14 15 16 17 18 19 20 21 22 23 24
1 0 1 2 3 4 5 6 7 8 9 10 11 12
500 650 700 800 850 900 950 1000 1050 1100 1150 1200
1 2 3 4 5 6 7 8 9 10 11 12
Eng ine Sp eed
K
Figure 3: This figure shows all possible grid points and cor-
responding index from which a 6× 6 grid must be chosen.
2.4.1 Choosing a Grid
The first constraint guarantees that we have exactly
p× p SS points in the grid:
∑
j∈S
x
j
= p× p (2)
Note that this constraint does not guarantee that all
of the points are chosen to be in the same rows and
columns. Figure 4 shows an example selection of 6×
6 = 36 highlighted points that were chosen based on
this constraint, but they do not form a 6× 6 grid.
10 450 109 110 111 112 113 114 115 116 117 118 119 120
9 400 97 98 99 100 101 102 103 104 105 106 107 108
8 350 85 86 87 88 89 90 91 92 93 94 95 96
7 300 73 74 75 76 77 78 79 80 81 82 83 84
6 250 61 62 63 64 65 66 67 68 69 70 71 72
5 200 49 50 51 52 53 54 55 56 57 58 59 60
4 150 37 38 39 40 41 42 43 44 45 46 47 48
3 100 25 26 27 28 29 30 31 32 33 34 35 36
2 50 13 14 15 16 17 18 19 20 21 22 23 24
1 0 1 2 3 4 5 6 7 8 9 10 11 12
0 500 650 700 800 850 900 950 1000 1050 1100 1150 1200
Figure 4: This figure shows highlighted in blue the 6× 6 =
36 points that were chosen to satisfy constraint (2).
The next two constraints address this problem and
guarantee that we have exactly p columns and p rows
chosen to compose the grid. Note that the two con-
straints (3) and (4) make constraint (2) redundant, but
having it improvesthe solve time by around 5% based
on our testing, as it gives a better representation of the
convex hull (typically, additional constraints in BIPs
help improve solve time (Geoffrion, 1976)). The con-
straint to ensure we choose exactly p columns is
∑
j
2
∈S:E
j
1
=E
j
2
x
j
2
= px
j
1
∀ j
1
∈ K. (3)
The following guarantees we choose exactly p rows:
∑
j
2
∈S:T
j
1
=T
j
2
x
j
2
= px
j
1
∀ k ∈ {0,. .. ,r − 1} (4)
∀ j
1
∈ S : j
1
= qk+ 1.
Figure 5 shows an example grid selection, choosing
columns 1, 2, 4, 7, 8, and 12 and rows 2, 4, 5, 7, 9, and
10. The next few constraints enforce maximum spac-
ing between the grid points in both the engine speed
and torque directions. We ensure that the first (i.e.,
left-most) column in the grid must be at most DX
max
away from the starting value of E
s
:
∑
j∈S:E
j
≤E
1
+DX
max
x
j
≥ p. (5)
10 450 109 110 111 112 113 114 115 116 117 118 119 120
9 400 97 98 99 100 101 102 103 104 105 106 107 108
8 350 85 86 87 88 89 90 91 92 93 94 95 96
7 300 73 74 75 76 77 78 79 80 81 82 83 84
6 250 61 62 63 64 65 66 67 68 69 70 71 72
5 200 49 50 51 52 53 54 55 56 57 58 59 60
4 150 37 38 39 40 41 42 43 44 45 46 47 48
3 100 25 26 27 28 29 30 31 32 33 34 35 36
2 50 13 14 15 16 17 18 19 20 21 22 23 24
1 0 1 2 3 4 5 6 7 8 9 10 11 12
0 500 650 700 800 850 900 950 1000 1050 1100 1150 1200
Figure 5: This figure an example grid with exactly 6 rows
and 6 columns.
Note that in Figure 5, column 1 is selected, and since
E
s
= 500, this constraint holds.
Next, we require the grid to be spaced at most
DY
max
in the torque direction:
x
j
1
+
∑
j
2
∈S:j
2
> j
1
∧|T
j
1
−T
j
2
|≤DY
max
x
j
2
≥ p+
∑
c∈S:T
c
=T
j
1
x
c
(6)
∀ k ∈ {0,. ..,((T
end
− DY
max
)/GY
min
)}
∀ j
1
∈ S : j
1
= qk+ 1.
Following Figure 5, we can see that the maxi-
mum space between any two rows is 100, and since
DY
max
= 100, this constraint holds.
Finally, we guarantee that the grid can be spaced
at most DX
max
in the engine speed direction:
x
j
1
+
∑
j
2
∈S:j
2
> j
1
∧|E
j
1
−E
j
2
|≤DX
max
x
j
2
≥ p+
∑
c∈S:E
c
=E
j
1
x
c
(7)
∀ j
1
∈ {1,.. .,((E
end
− DX
max
− E
s
)/GX
min
+ 1)}.
Following Figure 5, we can see that the maximum
space between any two columns is 200 (e.g. columns
8 and 12), and since DX
max
= 150, this constraint is
violated. Figure 6 shows a grid selection that satisfies
all constraints up to this point, with a maximum space
between any two columns equal to 150.
10 450 109 110 111 112 113 114 115 116 117 118 119 120
9 400 97 98 99 100 101 102 103 104 105 106 107 108
8 350 85 86 87 88 89 90 91 92 93 94 95 96
7 300 73 74 75 76 77 78 79 80 81 82 83 84
6 250 61 62 63 64 65 66 67 68 69 70 71 72
5 200 49 50 51 52 53 54 55 56 57 58 59 60
4 150 37 38 39 40 41 42 43 44 45 46 47 48
3 100 25 26 27 28 29 30 31 32 33 34 35 36
2 50 13 14 15 16 17 18 19 20 21 22 23 24
1 0 1 2 3 4 5 6 7 8 9 10 11 12
0 500 650 700 800 850 900 950 1000 1050 1100 1150 1200
Figure 6: This figure is an example grid selection that satis-
fies all constraints in Section 2.4.1.
2.4.2 Required Grid Points
There are a couple of types of SS points that are re-
quired to be a part of the grid. The first require-
ment forces the grid to contain points that have values
greater than or equal to the max torque on the FLC.
This is equivalent to forcing the maximum value in
the grid to be selected.
∑
j∈S:T
j
=T
e
x
j
= p. (8)
ICORES 2012 - 1st International Conference on Operations Research and Enterprise Systems
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