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Unformatted text preview: ated in a series of
lifts, each of which involves dumping just a few inches of soil and then compacting it. Deeper cuts
and higher ﬁlls require more work to be done on the road shoulders, and possibly, the addition of
125 reinforced concrete structures to stabilize the earthwork. This explains why the marginal cost of
cuts and ﬁlls increases with their depth/height.
We will work with a discrete model, specifying the road height as hi , i = 1, . . . , n, at points equally
spaced a distance d from each other along the given path. These are the variables to be chosen.
(The heights h1 , . . . , hn are called a grading plan.) We are given ei , i = 1, . . . , n, the existing
elevation, at the points. The grading cost is
n C=
i=1 φﬁll ((hi − ei )+ ) + φcut ((ei − hi )+ ) , where φﬁll and φcut are the ﬁll and cut cost functions, respectively, and (a)+ = max{a, 0}. The ﬁll
and cut functions are increasing and convex. The goal is to minimize the grading cost C .
The road height is constrained by given limits on the ﬁrst, second, and third derivatives:
hi+1 − hi /d ≤ D(1) , i = 1, . . . , n − 1 3 i = 3, . . . , n − 1, hi+1 − 2hi + h...
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This note was uploaded on 09/10/2013 for the course C 231 taught by Professor F.borrelli during the Fall '13 term at University of California, Berkeley.
 Fall '13
 F.Borrelli
 The Aeneid

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