Appendix. Online supplement to Coordinated logistics with a truck and a drone

Transcription

1 Appendix. Online supplement to Coordinated logistics with a truck and a drone

2 28 Article submitted to Management Science; manuscript no. MS R2 v 1 v 1 s s v 2 v 2 (a) (b) Figure 13 Reflecting the polygonal chain s about the line joining v 1 and v 2. A. Proof of Lemma 3 The assumption that L is polygonal is a valid one because the standard definition of the length of L is simply the limit of a discretization; if we say that L is the image of the continuous mapping γ : [0, 1] R 2, then the length of L is defined as length(l) = sup M1 0=t 0 <t 1 < <t M =1 γ(t i ) γ(t i1 ) where the supremum is taken over all possible partitions of [0, 1] and M is unbounded. The assumption that L forms the boundary of a convex region is also straightforward: at the very least, we can certainly assume that L is simple, that is, that L does not intersect itself (since one can always un-cross a pair of intersecting edges of L in an obvious way). If L is not the boundary of a convex region, then (since we have assumed that L is polygonal) there must exist a pair of vertices v 1, v 2 of L such that v 1 and v 2 are adjacent on the convex hull Conv(L) of L, but not adjacent on L itself; see Figure 13a. If we let s denote the component of L that lies between v 1 and v 2, then it is obvious that we can reflect s about the line joining v 1 and v 2, to obtain a new curve L with the same length as L, as shown in Figure 13b. It is then an entirely straightforward argument to verify that, for any ɛ, we have Area(N ɛ (L )) Area(N ɛ (L)), which completes the proof. B. Zig-zagging and spiralling Here we give a precise description of the construction of a zig-zag or spiral tour for sufficiently large l, where we assume that R is convex with area A. We describe two schemes in which one is given a length l and gives a loop L such that length(l) l and such that R d(x, L) dx A2 /(4l) as l. Zig-zagging We can represent R as the region bounded by a convex function and a concave function of one variable, i.e. R = {x = (x 1, x 2 ) R 2 : f(x 1 ) x 2 g(x 1 )} for some convex f and concave g, whence Area(R) = A = b g(t) f(t) dt for appropriate horizontal bounds a and b. Assume without loss of generality a that a = 0 and b = 1, so that Riemann integration yields A = g(t) f(t) dt = lim N N where we define l i := g(i/n) f(i/n). Let s i 1 g(i/n) f(i/n) = lim N N l i, denote the line segment that corresponds to l i, and note that (provided that N is even) we can join the s i s together to form a loop L whose length is at most

3 Article submitted to Management Science; manuscript no. MS R2 29 Figure 14 (a) (b) Joining a family of Riemann rectangles (a) into a loop (b). N l i + 2P, where P is the perimeter of R (a constant); Figure 14 shows how this is done. As N, the above can be equivalently written as N l i AN. Further note that the distance between a uniformly selected point x R and its nearest segment s i converges to a uniform distribution between 0 and 1/(2N), which establishes that the expected distance between uniform x and its nearest s i converges to 1/(4N), whence d(x, L) dx A/(4N) as N. Therefore, given l, we can set N to be the largest even integer R such that N l i + 2P l, and then form L by joining the appropriate segments s i. We see that as l, we have N as well, whence l l i AN = as desired. R d(x, L) dx A 4N A 4l/A = A2 4l Spiralling It will suffice to build a family of concentric loops L = {L i } within R such that the desired conditions hold, i.e. R min i d(x, L i ) dx A 2 /(4l). This is sufficient because we can stitch these loops to form a single large loop L by taking convex combinations of consecutive loops, as suggested in Figure 15. This process does not increase the total length by more than the diameter D of R, that is, length(l) length(l i i) + D. This construction is similar to the proof of Lemma 3. For any ɛ, let L ɛ be the loop consisting of all points x R that are a distance of exactly ɛ away from the boundary of R (which would itself therefore be written as L 0 ). The coarea formula (Krantz and Parks 2008) says that the area A of R can be written as A = ɛ 0 length(l ɛ ) dɛ where ɛ is the radius of R, i.e. ɛ = max x R d(x, L 0 ). Assume without loss of generality that ɛ = 1, so that Riemann integration yields A = length(l ɛ ) dɛ = lim N N 1 length(l i/n ) = lim N N l i, where we define l i := length(l i/n ). As N, the above can equivalently be written as N l i AN. Further note that the distance between a uniformly selected point x R and its nearest loop L i converges to a uniform distribution between 0 and 1/(2N), which establishes that the expected distance between uniform

4 30 Article submitted to Management Science; manuscript no. MS R2 Figure 15 (a) (b) Joining a family of concentric loops (a) into a single loop (b). x and its nearest L i converges to 1/(4N), whence R min i d(x, L i ) dx A/(4N) as N. Therefore, given l, we can set N to be the largest even integer such that N l i + D l, and then form L by joining consecutive concentric loops L i. We see that as l, we have N as well, whence as desired. l C. Proof of Theorem 9 l i AN = R d(x, L) dx A 4N A 4l/A = A2 4l Suppose that L is a loop with length l := c n (where we would have c = 0 /(2 1 ) in Theorem 9) and suppose that points Q 1,..., Q n are uniformly distributed (with respect to the arc length measure) on L. If we select a point Q i arbitrarily, we see that the clockwise distance from Q i to any other point Q j is uniformly distributed between 0 and l. Therefore, the distance D from Q i direction has a cdf F (d) given by ( ) n1 l d F (d) = 1. l to its nearest neighbor in the clockwise We want to show that D can be approximated by an exponential distribution with rate parameter λ = n/c; more precisely, we will show that the random variable D n converges in distribution to an exponential distribution with rate parameter 1/c: as desired. lim Pr(D n d) = lim Pr(D d/ n) = lim F (d/ n) = lim 1 ( ) n1 ( ) n1 c n d/ n cn d c = lim n 1 = 1 e d/c cn C.1. A joint distribution We can use an entirely analogous argument to establish that the distances between two consecutive points along L can be approximated with a pair of independent exponential random variables. Again, as in the

5 Article submitted to Management Science; manuscript no. MS R2 31 preceding argument, we know that the distance D 1 from an arbitrarily selected point Q i to its nearest neighbor Q j in the clockwise direction has a cdf F 1 (d 1 ) given by ( ) n1 l d1 F 1 (d 1 ) = 1, l and it is also easy to see that the distance D 2 from Q j conditioned on the event that D 1 = d 1, has a cdf F 2 (d 2 D 1 = d 1 ) given by ( ) n2 l d1 d 2 F 2 (d 2 D 1 = d 1 ) = 1. l d 1 to its nearest neighbor in the clockwise direction, By differentiating, we see that the corresponding probability density functions are given by f 1 (d 1 ) = n 1 (l d 1) n2 f 2 (d 2 D 1 = d 1 ) = which therefore gives a joint cdf F (d 1, d 2 ) given by F (d 1, d 2 ) = d1 d2 0 0 l n1 n 2 (l d 1 ) n2 (l d 1 d 2 ) n3 f 1 (d 1)f 2 (d 2 D 1 = d 1) dd 2 dd 1 d1 d2 (n 1)(n 2) = (l d l n1 1 d 2 ) n3 dd 2 dd ( ) n1 ( ) n1 ( l d1 d 2 l d1 l d2 = 1 + l l l We claim that the joint distribution on D 1 n and D2 n converges to a joint cdf on two independent exponential random variables with rate parameter 1/c: as desired. lim Pr(D 1 n d1 D 2 n d2 ) = lim Pr(D 1 d 1 / n D 2 d 2 / n) = lim F (d 1 / n, d 2 / n) = lim 1 + = lim 1 + ) n1 ( c n d1 / n d 2 / ) n1 ( n c n c d1 / ) n1 ( n c n n c d2 / n n c n ( ) n1 ( ) n1 ( ) n1 cn d1 d 2 cn d2 cn d1 cn cn cn =1 + e (d 1+d 2 )/c e d 1/c e d 2/c = ( 1 e d 1/c ) ( 1 e d 2/c ) D. Proof of Theorem 10 Assume (purely for notational purposes) that the optimal permutation σ is the identity and consider the sequence of stops traversed by the UAV: x 1 p 1 y 1 x 2 p 2 y 2 x n p n y n x 1. Let l 0 = n y i x i+1 and l 1 = n x i p i + p i y i, so that l 0 represents the length that is traversed jointly with the truck and the UAV and l 1 represents the length that is traversed exclusively by ) n1

6 32 Article submitted to Management Science; manuscript no. MS R2 the UAV. Our battery life constraint says that l 1 1 τ. Certainly, we have l 0 + l 1 TSP, and therefore the optimal duration OPT is bounded below by the linear program 1 minimize l l 1 l 0,l s.t. l 0 + l 1 TSP l 1 1 τ. Using the fact that 0 < 1, standard arguments guarantee that l 0 = TSP 1 τ and l 1 = 1 τ, which tells us that OPT 1 0 TSP ( 1 / 0 1)τ for all τ. To prove that equality holds when τ is small, we simply have the truck traverse the TSP tour of the p i s, and then cut corners with the UAV when the TSP tour makes an angle that is sufficiently acute (the condition that there exists θ such that θ 2 arcsin( 0 / 1 ) merely asserts that there exists one such angle). E. Raw data from Section 4.2

7 Article submitted to Management Science; manuscript no. MS R2 33 n = 25 n = 50 n = = 30 km 1 = 40 km 1 = 50 km 1 = 60 km T P H P/H T P H P/H T P H P/H T P H P/H Table 3 The above table is divided into 3 4 blocks, each corresponding to a choice of n and 1. Each block contains 10 rows, each corresponding to a single experiment, and 4 columns labelled T, P, H, and P/H, which ar e interpreted as follows: column T is the duration, in hours, of a TSP tour of the n points when using road network distances and times as determined by the Google Distance Matrix API (Developers 2015) and the Concorde TSP Solver (Concorde 2015). Column P is the predicted duration of a horsefly tour of the n points, as predicted from equation (18). Column H is the actual duration of a horsefly tour of the n points, again using the the Google Maps Directions API (Google Developers 2015), the Google Distance Matrix API (Developers 2015), and the 2-and 3-OPT heuristic described in the previous section. The percent error of the prediction is given in the fourth column, which is labelled P/H, and shows the ratio of the predicted horsefly tour length to the actual horsefly tour length.

8 34 Article submitted to Management Science; manuscript no. MS R2 n = 25 n = 50 n = = 30 km 1 = 40 km 1 = 50 km 1 = 60 km T P H P/H T P H P/H T P H P/H T P H P/H Table 4 Results of computational experiments where the truck is permitted to visit destination points. All data fields are the same as in Table 3.

9 Article submitted to Management Science; manuscript no. MS R2 35 n = 25 n = 50 n = = 30 km 1 = 40 km 1 = 50 km 1 = 60 km T P H P/H T P H P/H T P H P/H T P H P/H Table 5 Results of computational experiments when UAV battery life is limited to a half hour, i.e. τ = 0.5. All data fields are the same as in Table 3.

10 36 Article submitted to Management Science; manuscript no. MS R2 n = 25 n = 50 n = = 30 km 1 = 40 km 1 = 50 km 1 = 60 km T P H P/H T P H P/H T P H P/H T P H P/H Table 6 Results of computational experiments when UAV battery life is limited to an hour, i.e. τ = 1. All data fields are the same as in Table 3.

11 Article submitted to Management Science; manuscript no. MS R2 37 n = 25 n = 50 n = = 30 km 1 = 40 km 1 = 50 km 1 = 60 km T P H P/H T P H P/H T P H P/H T P H P/H Table 7 Results of computational experiments when UAV battery life is limited to two hours, i.e. τ = 2. All data fields are the same as in Table 3.