Multi-Stage Recoverable Robustness

the latter using δ≤ ^α₂. Hence,

X

u∈V

(xu−πu) =δ,

i.e. π is robust. The price of robustness follows from Lemma 6.21.

6.4 Multi-Stage Recoverable Robustness

Now we extend the single-stage model in order to deal with a sequence of σ ≥ 1 modifications. The main motivation lies on the observation that in many applications, one is typically not facing only one disturbing event, but several disturbancesi¹, i², . . . , i^σ may occur. This obviously is the case for the operational phase in public transportation where several different delays might occur one after another. For example, assume that we expect at most two disturbances i¹ andi². In this case, a robust solution for i¹ should be also recoverable against the next disturbance i². This means that under all solutions which are robust fori¹, we should choose one that is again robust against the next disturbancei² (if it exists). This example can be extended to more than two disturbances, see Figure 6.5 for an example where up to 3 disturbances are allowed.

F(i¹) F(i²) F(i³)

F(i⁰)

F(i⁰,3) F(i⁰,2) F(i⁰,1)

F(i¹,2) F(i¹,1)

F(i²,1)

Figure 6.5: The set of solutions that are recoverable against 1, 2, and 3 disturbances.

F(i, n) denotes the set of feasible solutions for a problem which has to be solved against n disturbances. Dotted arrows represent recovery algorithms.

To describe robustness, as for the single-stage case, the following concepts are necessary.

• σ ∈N denotes the maximum number of expected modifications. In a practical scenario, several disruptions i¹, i², . . . , i^σ may occur. In this case, the task is to devise recovery algorithms that can recompute the solution forP aftereach disruption. The introduction of more than one disturbance extends the concept of recoverable robustness presented in Section 6.2 where only one single disturbance is taken into account.

• The modification functionM : I →2^I. Note that M might also depend on other information, e.g. M(i^k) may depend on data of instances i⁰,i¹,. . .,i^k−1.

• Again, Arec denotes the class of recovery algorithms forP. Note thats⁰ and i¹ define the minimal amount of information necessary to recompute the recovered solution. However, for specific cases,Arec could require additional information.

In general, whenA_rec is used in thek-th stage, it could use everything that has been processed in the previous stages, i.e. i⁰, ..., i^k−1 and s⁰, ..., s^k−1.

The following definitions are extensions of Definition 6.1 and Definition 6.2 to the multi-stage case:

Definition 6.23. A multi-stage recoverable robustness problem is given by a tuple (P, M,Arec, σ) where(P, M,Arec)∈RRPis a recoverable robustness problem andσ∈N. The class RRP(σ) contains all multi-stage recoverable robustness problems, i.e. all recoverable robustness problems that have to be solved againstσ ≥1 possible disruptions.

Definition 6.24. Letσ ∈N and P = (P, M,Arec, σ) be an element of RRP(σ). Given an instance i⁰∈I of P, s⁰ is a feasible solution for i⁰ with respect to P if and only if the following relationship holds:

∃A_rec ∈Arec: s⁰∈F(i⁰) (6.10)

s^k:=Arec(s^k−1, i^k)∈F(i^k) ∀i^k∈M(i^k−1), ∀k∈ {1, . . . , σ}. (6.11) This definition ensures that for each stage k, for any possible modificationi^k∈M(i^k−1) and for any feasible solution s^k−1 computed in the previous stage, the output s^k of algorithm A_rec is a feasible solution for i^k with respect to P. If it is clear to which problemP,M andArec we refer to, we also say in short thats⁰ is feasible fori with respect to σ recoveries. As in the single-stage case, F_P(i) is considered as the set of robust solutions for iwith respect to the original problem P.

Note that RRP(1) = RRP. Hence, each problem in RRP(1) is called a single-stage recoverable robustness problem and each problem in RRP(σ),σ >1, is called a multi-stage recoverable robustness problem.

Using the definition of a feasible solution, the robust algorithm that is used to compute the initial solutions⁰ for the initial (undisturbed) instancei⁰ is defined in the following definition similar to Definition 6.3 for the single-stage case:

Definition 6.25. Given a multi-stage recoverable robustness problemP ∈RRP(σ), a robust algorithmfor P is any algorithm Arob :I →S such that for each i∈I, Arob(i) is feasible for i with respect to P, i.e. such that A_rob outputs a solution that can be recovered against σ disturbances.

6.4 Multi-Stage Recoverable Robustness

Note that A_rob as in the definition above provides the solution s⁰ as defined in Def-inition 6.24. In the case of strict robustness, a robust algorithm Arob for P must provide a solutions⁰ fori⁰ such that for each possible modification i^k∈M(i^k−1), we haves⁰ ∈F_P(i^k) for allk∈ {1, . . . , σ}. The meaning is the following: If A_rec has no recovery capability, thenArob has to find solutions that “absorb” any possible sequence of disturbances.

Analogously to the single-stage case, the price of robustness can also be defined for multi-stage recovery algorithms. Definitions 6.26-6.28 are generalizations of Definitions 6.4-6.6 to the multi-stage case.

For every instancei∈I of the initial problem P, the price of robustness of the robust algorithmA_rob is given by the maximum ratio between the cost of the solution provided by A_rob and the cost of an optimal (non-robust) solution. The following definition differs from the corresponding definition for the single-stage case only in the problemP (it now belongs to the larger class RRP(σ) instead of RRP).

Definition 6.26. The price of robustness of a robust algorithm A_rob for a recoverable robustness problem P ∈RRP(σ) is given by

P_rob(P, A_rob) := max

i∈I

f(A_rob(i)) min{f(x) :x∈F(i)}

.

The price of robustness of a recoverable robustness problem P ∈RRP(σ) is then given by the minimum price of robustness among all possible robust algorithms for this problem. Formally:

Definition 6.27. Theprice of robustnessof a multi-stage recoverable robustness problem P ∈RRP(σ) is given by

P_rob(P) = min{P_rob(P, A_rob) :A_rob is a robust algorithm for P}.

If there are different robust algorithms for a recoverable robustness problem, we want to identify the “best” one:

Definition 6.28. Let P ∈RRP(σ) and let A_rob be a robust algorithm for P. Then,

• A_rob is called P-optimalif P_rob(P, A_rob) =P_rob(P);

• Arob is called exactif Prob(P, Arob) = 1.

We call a solution computed by an optimal algorithm P-optimal, while a solution computed by an exact algorithm is called exact.

Note that each exact algorithm is optimal.

We now state a simple observation concerning the price of robustness.

Lemma 6.29. For fixed P, M, and Arec, consider a family of recoverable robustness problemsP_σ = (P, M,Arec, σ)∈RRP(σ) for different values of σ, i.e. these problems vary in the expected number of recoveries only. For σ1 < σ2, we have

• FPσ2(i)⊆FPσ1(i) for all instances i∈I,

• P_rob(P_σ1)≤P_rob(P_σ2), i.e. the price of robustness grows in the number of expected recoveries.

Proof. Let Arob be a robust algorithm for P_σ2. Let i ∈ I and s ∈ FPσ2(i). By Definition 6.24, there exists a recovery algorithm Arec ∈Arec such that (6.11) holds for allk= 1, . . . , σ₂, hence also for all k= 1, . . . , σ₁ < σ₂. This impliess∈FPσ1(i).

The second statement is a straightforward consequence of the first one.