Let ω(r) = max
T:T⊆Zℓ
i∈T
d(i) P
i∈Npi·r(T)
pi. We see that ω(r) is a non-increasing function of r. Therefore, the solution to the program (4.1) is the same as the solution of the equation r=ω(r), which is max
T:T⊆Zℓ
ωT. 2
is achieved when fki =r1, fkj =r1.
Assume Woff-b is of order r1. Since r2 ≫ r1, the non-broken vehicles outside the circle can’t move to positionsiorj. The only remaining non-broken vehicle is k. The vehicle k needs to walk back and forth between i and j because the service requests arrive at positions i and j alternatively. In order to serve all the requests, the travel distance ofkisr1+(2r1−1)·2r1, which is not of orderr1. A contradiction. Therefore, Woff-b is not of order r1 and Woff-b=ω(r1).
From this example, we see that, when the vehicle breaking down is allowed, the job arriving order matters and the bound given in Section 4.1 is a weak lower bound, which is not tight up to a constant factor.
Chapter 5
Inter-Vehicle Energy Transfers
In Chapters 2 and 3, we discussed the problem of minimizing the initial energy needed for each vehicle in the off-line and on-line cases respectively. We now extend the problem to allow inter-vehicle energy transfers. That is, vehicleAcan transfer energy to vehicle B when A and B are at the same location. We consider two methods of accounting for energy transfers:
• Fixed cost: Chargea1 units of energy per transfer, no matter how much energy is transferred, or
• Variable cost: Charge a2 ≪1units of energy per unit of energy transferred.
It turns out that, no matter which accounting method is used, under the same job load, the minimal vehicle capacity W needed with energy transfer allowed is of the same order as that without energy transfer. We let Wtrans-off andWtrans-on denote the minimalW in the off-line and on-line energy-transfer cases respectively. Clearly,
Wtrans-off ≤Wtrans-on, Wtrans-off ≤Woff, and Wtrans-on ≤Won,
so it is enough to show that Wtrans-off = Θ(Woff). This, together with Theorem 1.4.2, shows that all fourW’s (Wtrans-off,Wtrans-on,Woff, andWon) are of the same order. In the following, we will first show that Wtrans-off = Θ(Woff). Then we provide a brief discussion on how high capacity tanks can help performance.
In this section, the analysis in the proof of Theorem 5.1.1 gives a lower bound on Wtrans-off. It works for both energy-transfer accounting methods: the fixed cost sce- nario where each transfer costs a1 units regardless of amount transferred, and the variable cost scenario where the transfer costs a2 units per unit of energy transferred.
Once again, for simplicity, we only study the two-dimensional (ℓ= 2) case.
Theorem 5.1.1 Wtrans-off = Θ(Woff).
Proof : Since there is at mostWtrans-off units of energy in each vehicle, the minimum traveling cost per unit energy per distance is W 1
trans-off
1. Moving Wtrans-off units of energy from a position i to another position j will cost energy at least
Wtrans-off× 1
Wtrans-off
+Wtrans-off
1− 1 Wtrans-off
× 1
Wtrans-off
+· · ·+Wtrans-off
1− 1 Wtrans-off
ki−jk−1
× 1
Wtrans-off
= Wtrans-off× 1−
1− 1 Wtrans-off
ki−jk! ,
and so the left energy amount is at most Wtrans-off ×
1− W 1
trans-off
ki−jk
, no matter how many transfers of energy occur, nor which transfer accounting method is used.
Therefore, when moving Wtrans-off units of energy from position i into an s ×s square T ⊆Z2, the amount of remaining energy, which is initially fromi, is at most
Wtrans-off×
1− 1 Wtrans-off
D(i,T)
,
where D(i, T) is the Manhattan distance from the point i to the squareT. The amount of energy initially in the square T isWtrans-off×s2. Since
|{i:D(i, T) =r}|= 4s+ 4(r−1),
1The traveling cost is 1 unit of energy per unit of distance.
the total amount of energy that can be moved into the square T is at most
Wtrans-off×s2+ X∞
r=1
Wtrans-off×(1− 1
Wtrans-off
)r×(4s+ 4(r−1))
= Wtrans-off× s2+ 4 X∞
r=1
(1− 1
Wtrans-off)r×(r+s−1)
!
= Wtrans-off× s2+ 4 X∞
r=1
r(1− 1 Wtrans-off
)r+ 4(s−1) X∞
r=1
(1− 1 Wtrans-off
)r
!
= Wtrans-off× s2+ 4(Wtrans-off−1)Wtrans-off+ 4(s−1)(Wtrans-off−1)
= Wtrans-off× s2+ 4Wtrans-off2 + 4sWtrans-off−8Wtrans-off−4s+ 4 ,
which should be at least the total demand in T. That is,
Wtrans-off× s2+ 4Wtrans-off2 + 4sWtrans-off−8Wtrans-off−4s+ 4
≥X
i∈T
d(i).
Since
|NWtrans-off(T)|
= Θ s2+Wtrans-off2
= Θ s2+ 4Wtrans-off2 + 4sWtrans-off−8Wtrans-off −4s+ 4 ,
we have that Wtrans-off = Ω (ωT), where ωT denotes the solution of ωT · |NωT(T)| = P
x∈T d(x). Let Γ be the set of all squares in Z2. ThenWtrans-off = Ω
maxT∈ΓωT
. By Corollary 2.2.6, Woff = Θ
maxT∈Γ ωT
. Since Wtrans-off ≤ Woff, we have that Wtrans-off = Θ
maxT∈Γ ωT
= Θ(Woff). 2
Corollary 5.1.2 Wtrans-off, Wtrans-on, Woff, and Won are Θ
maxT ωT
.
Often in practical applications, the vehicle tanks might not be full initially. Let W denote the initial energy andC > W denote the vehicle tank capacity. When energy transfer is not allowed, the minimal W is the same as discussed in Chapters 2 and 3 regardless of C. However, when energy transfer is allowed, large capacity tanks can improve performance. We present an interesting example where Wtrans−off = Θ (avgxd(x)).
5.2.1 An example of W
trans−offwith C = ∞
For this illustration, one dimension suffices. Consider a line segment of length N (vertices numbered 1 through N). When the vehicle capacities are infinite, we let vehicle-1 travel toN while collecting energy from vehicles2,3, . . . , N−1along the way.
AtN, it exchanges energy with vehicleN such that vehicle-N has exactly the amount of energy required to process the jobs at its position. After the exchange, vehicle-1 walks back to its original position while distributing energy to vehicles N−1, . . . ,3,2 according to the demand required at each position. In total, the number of energy transfers is2N −3 and the distance traveled is 2N −2.
• In the fixed cost scenario where each transfer costsa1 units regardless of amount transferred, the total energy needed is
Etotal=a1·(2N −3) + (2N −2) +X
x
d(x).
Then
Wtrans−off = a1 ·(2N −3) + (2N −2) +P
xd(x) N
= 2a1 + 2 + P
xd(x)−3a1−2
N .
• In the variable cost scenario where the transfer costsa2 units per unit of energy
transferred, the total energy needed is
Etotal =a2·Wtrans−off·(2N−3) + (2N −2) +X
x
d(x).
Then
Wtrans−off ·N =a2·Wtrans−off·(2N−3) + (2N −2) +X
x
d(x)
⇒ Wtrans−off = 2N −2 +P
xd(x) N −2a2N + 3a2
.
Therefore, Wtrans−off = Θ (P
xd(x)/N) = Θ (avgxd(x)), no matter which energy- transfer accounting method is used.
From this example, we see that there is significant performance difference between the C =W and C 6=W in the off-line case.
Chapter 6
Conclusions and Future Works
The key concepts/contributions of the thesis is as follows:
• The minimal energy needed in the off-line case is of the same order as that of the on-line case.
• We proposed an algorithm and a strategy for the Capacitated Multivehicle Routing Problem.
We note that the proposed algorithm represents only an initial prototype. We expect that this thesis will provide an incentive for future work on this problem. There are also several immediate research directions that can be pursued:
• The example from Section 5.2.1 shows that when inter-vehicle energy transfer is allowed and vehicles posses (non-full) large-capacity tanks, there can be sig- nificant energy savings. The questions of how much energy could be saved in general remains open.
• One can explore further the precise relationship between Woff and Won, trying to tighten the constant factor, which is exponential inℓ. In particular, it would be nice to show that the exponential dependence on ℓ is unnecessary.
• We have only discussed the case where the underlining graph is a grid. It would be nice to have results for graphs in general.
Ongoing advances in technology suggests that electronic devices will become smaller, smarter, more mobile, and ubiquitous. We hope to see the applications of these de- vices benefit from the theoretical research of CMVRP. The rich applications of such mobile devices will surely yield many interesting theoretical research problems as well.
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