Chapter 5. Deployment Scheduling
5.3. Resource Delivery Path Dependent Deployment Scheduling
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work teams. Each node must be fixed before its descendants. The construction sequence is called CCN-DS problem. We proposed a mathematical model to formulate CCN-DS problem and use CCN-DS algorithm to find a deployment schedule to maximize the efficiency of disaster response operation [27]. However, CCN-DS didn’t take the traveling time of each selected path into account. This research is trying to improve the CCN-DS model by taking traveling time into account.
5.3. Resource Delivery Path Dependent Deployment Scheduling
The emergency level of the area covered by a base station is represented by a time-variant profit parameter, which has to be defined by the disaster response authority because they have not only the necessary knowledge for disaster response, but also the official authority as well as situation statistics. A typical example of profit definition is the estimated time dependent survival rate.
The Resource Delivery Path Dependent CCN Deployment Scheduling Problem is formulated into two optimization models: the first one, CCNDS-AC, is with Antecessor Precedence Constraint, and the second one, CCNDS-UC, is without the constraint. We assume one or more forwarding tree is calculated in advance.
The non-preemptive CCNDS-AC is as follows. A set of nodes organized in a tree structure has to be fixed by work teams; a preceding node must be rescued before its descendants; the profit of fixing a node is a function of time; the traveling time from each node to every other node is also given; the CCNDS-AC problem is to find a deployment sequence such that the total profit is maximized. An example is shown in Fig. 5.1.
The graph on the left of Fig. 5.1 is the forwarding tree. The blue edges are the wireless links
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and the red paths are traveling paths labeled with traveling time. The table on the right shows the deployment sequence labeled with traveling time. The value in each table cell is a time-variant profit.
Mathematical model of CCNDS-AC is as follows.
Figure 5.1. Example of deployment schedule
The input parameter are G’(V,E), R, π, D, P that are defined as follows. The input graph, G’, is the network topology CCN.
V = {vi | i=0,1,2,…,n}, is the set of base stations and v0 is the root which has an external link.
E = {ei,j | ei,j , vi,vj ϵ V}is the set of link.
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R = {ri | i=1,2,…,n and ri >0} is the set of construction time of the isolated station vi.
π={πa | πa = (πa(1),…,πa(n)) is the set of CCN deployment schedules, where a is a positive integer. π = {πi|(π(1), … , π(i)), i = 1, … , n}. πa(i) is the position of vi in schedule πa.
D={di,j | vi,vj ϵ V} is the set of traveling time.
P = {Pi(t) | vi ϵ V} is the set of profit, Pi(t) is the profit of vi when it is constructed at time t and with respect to a schedule πa , the construction time of vi is ∑ rπa(k) where k is a node precedes node vi in schedule πa.
The CCN deployment scheduling problem is to find a deployment schedule πa from π, to Maximize
∑ Pi(t) = ∑ Pi(c(πa,i)), i=1,2,…,n Subject to
πa(i) precedes πa(j), where vj is the descendant of vi for all vi,vj ϵ V.
For the sake of discussion, an isolated node is defined as a node whose parent node hasn’t been rescued (visited). The antecessor precedence constraint in CCNDS-AC forces the transportation vehicle to ignore any isolated node even if the vehicle passes such a node.
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However, it might be beneficial to visit such an isolated node before its preceding nodes.
Even though a rescued isolated node is not able to provide any service immediately after it is rescued, it can be activated without an extra rescue trip immediately after its parent node is rescued. Taking the example shown in Fig. 5.2, the red path (A,B,E) in the left graph is a solution of CCNDS-AC and the path (E,B,A) in the right graph is a solution ignoring antecessor precedence constraint. Both paths started from the headquarter that is located near node E. As we can see that the traveling time of the path on the right graph is smaller than its counterpart on the left graph. Take this consideration, we propose another model, Unconstrained CCN Deployment Scheduling Problem (CCNDS-UC) which is the same as CCNDS-AC but ignoring antecessor precedence constraint.
Figure 5.2. A path ignoring antecessor precedence constraint.
5.4. Complexity Analysis
Similar to the conventional machine scheduling problem, CCNDS-AC can be easily proven NP-hard.
CCNDS-AC is in NP：
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We first show that CCNDS-AC ϵ NP. Assuming that we are given a forwarding tree T(V,E), as well as a schedule, we can use a double loop to verify that a parent node must be visited before its child nodes in T. The verification algorithm can affirm that the schedule is a valid CCNDS-AC schedule within O(n2) time.
CCNDS-AC is NP-Hard：
We now prove that CCNDS-AC problem can be reduced to the single machine scheduling problem (SMS) straightforwardly.
SMS is defined as follows: Given a set 𝐽 of n independent jobs that has to be scheduled on a single machine. Each job 𝑗𝑖 ∈ 𝐽 contains uninterrupted processing time 𝑢𝑖 ∈ 𝑈 and weight 𝑤𝑖 ∈ 𝑊, where 𝑢𝑖 and 𝑤𝑖 are positive integers. The single machine can handle only one job at a time.
SMS is to find a schedule π such that ∑𝑛𝑖=1(𝑤𝑖∗ 𝐶𝑖( 𝜋 )) is minimized, where π is a permutation of all jobs (𝑘 = 1, 2, 3, … , 𝑛!), 𝐶𝑖(𝜋) is the time at which job 𝑗𝑖 completes in the given schedule π.
Given an instance A:[J,W,U] in SMS, we can find an instance B:[V,E,R,D,P] with a single-level forwarding tree in CCNDS-AC such that an optimal solution πb for B is also an optimal solution for A. Let V=J, D={0|all paths}, R=U, P={−𝑤𝑖𝑡 , | 𝑤𝑖 ∈ 𝑊 }, E={eroot,i|vroot,vi ϵ V}. The verification can be performed in polynomial time. Let total weighted completion time of π for SMS is TWC(π) = ∑𝑛𝑖=1(𝑤𝑖∗ 𝐶𝑖(𝜋)), total weighted profit of π for CCNDS-AC is TWP(π) = −TWC(𝜋). We prove the following 3 Lemmas first：
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Lemma 1: Any valid schedule πb for B is a valid solution for A.
Proof: Any permutation of J is a valid schedule for A, and πb is a permutation of V, which is equal to J. Therefore πb is a valid solution for A. Q.E.D.
Lemma 2: Any valid schedule πa for A is also a valid schedule for B.
Proof: Any given schedule πa for A is a permutation of J which is equal to V. Therefore, πa is a permutation of V. Since each node in B can directly connect to the root of B such that the ancestor precedence constraint is always non-existing. Therefore, πa, a permutation of V, is a valid schedule for B. Q.E.D.
Lemma 3: If TWC(πa) < TWC(πb), then TWP(πa) > TWP(πb).
Proof: If ∑iϵJ(wi*Ci(πa)) < ∑iϵJ(wi*Ci(πb)), by Equal Division Theorem, we can get
∑iϵN(wi/Ci(πa)) > ∑iϵN(wi/Ci(πb)). Q.E.D.
Next, we prove by contradiction that an optimal solution πbfor B must be an optimal solution for A. By Lemma 1, we know πb is also a valid schedule for A, whose total weighted completion time is TWC(πb). Assume πb is not an optimal schedule for A, there must be another schedule πa, whose total weighted completion time TWC(πa) is smaller than TWC(πb).
By Lemma 2, πa is also a valid schedule for B, whose total weighted profit is TWP(πa). By Lemma 3, we can obtain TWP(πa) is greater than TWP(πb). This contradicts to the fact that πb
is an optimal solution for B. Therefore, πb must be an optimal solution for A. Q.E.D.
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Similarly, CCNDS-UC can also be proven to NP-hard in a similar way.