Analysis of budget for interdiction on multicommodity network flows

Abstract

In this paper, we concentrate on computing several critical budgets for interdiction of the multicommodity network flows, and studying the interdiction effects of the changes on budget. More specifically, we first propose general interdiction models of the multicommodity flow problem, with consideration of both node and arc removals and decrease of their capacities. Then, to perform the vulnerability analysis of networks, we define the function F(R) as the minimum amount of unsatisfied demands in the resulted network after worst-case interdiction with budget R. Specifically, we study the properties of function F(R), and find the critical budget values, such as $R_a$, the largest value under which all demands can still be satisfied in the resulted network even under the worst-case interdiction, and $R_b$, the least value under which the worst-case interdiction can make none of the demands be satisfied. We prove that the critical budget $R_b$ for completely destroying the network is not related to arc or node capacities, and supply or demand amounts, but it is related to the network topology, the sets of source and destination nodes, and interdiction costs on each node and arc. We also observe that the critical budget $R_a$ is related to all of these parameters of the network. Additionally, we present formulations to estimate both $R_a$ and $R_b$. For the effects of budget increasing, we present the conditions under which there would be extra capabilities to interdict more arcs or nodes with increased budget, and also under which the increased budget has no effects for the interdictor. To verify these results and conclusions, numerical experiments on 12 networks with different numbers of commodities are performed.

Appendices

Appendix 1: Proof of Proposition 1

Proof

By the construction process of network $G'$ from G, the objectives in (2) are the same for both networks, as $c'_{i_1,i_2}=0$ for any $i\in N$ and $c'_{i_2,j_1}=c_{ij}$ for any $(i,j)\in A$.

Let ${z'}_{i_1,i_2}^k$ for $i\in A$ and ${z'}_{i_2,j_1}^k$ for $(i,j)\in A$ be commodity flow of type $k\in K$ in network $G'$. Similarly, let ${y'}_{i_1,i_2}^k$ for $i\in A$ and ${y'}_{i_2,j_1}^k$ for $(i,j)\in A$ be corresponding interdiction variables on arcs of $G'$, and let $\mathbf y'$ denote the vector of all these interdiction variables, which is limited by ${\mathscr {I}}$ with the budget R and interdiction costs $q'_{i_1,i_2},q'_{i_2,j_1}$ are only on arcs. The budget constraint $\sum _{i\in N}p_{i}x_i+\sum _{(i,j)\in A} q_{ij}y_{ij}\le R$ becomes $\sum _{(i,j)\in A'} q'_{ij}y'_{ij}\le R$.

Constraints (2b) and (2e) are equivalent for both networks G and $G'$ (see [32] for explanations). In model (2), except the source and destination nodes, node-capacity constraints $\sum _{j:(i,j)\in A} \sum _{k\in K} z_{ij}^k \le v_i(1-x_i)$ can be equivalently transformed into arc-capacity constraints $\sum _{k\in K} {z'}_{i_1, i_2 }^k \le u^\prime _{i_1,i_2}(1-y^\prime _{i_1,i_2})$ in $G'$, while constraints $\sum _{k\in K} z_{ij}^k \le u_{ij}(1-y_{ij})$ in G become $\sum _{k\in K} {z'}_{i_2, j_1}^k \le u'_{i_2, j_1}(1-y'_{i_2, j_1})$ in $G'$. Additionally, node-capacity constraint $\sum _{j:(i,j)\in A} \sum _{k\in K} z_{ij}^k \le v_i(1-x_i)$ for source node $i\in G$ is equivalent to $\sum _{k\in K} {z'}_{i_1, i_2 }^k \le {u'}_{i_1,i_2}(1-{y'}_{i_1,i_2})$ on arc $(i_1,i_2)$ in $G'$, where ${u'}_{i_1,i_2}=v_i$. Node-capacity constraint $\sum _{j:(i,j)\in A} \sum _{k\in K} z_{ij}^k \le v_i(1-x_i)$ for destination node $i\in G$ now becomes $\sum _{k\in K} {z'}_{i_1, i_2 }^k \le {u'}_{i_1,i_2}(1-{y'}_{i_1,i_2})$ on arc $(i_1,i_2)$ in $G'$, where ${u'}_{i_1,i_2}=v_i$. Thus, the node interdiction on i in G is equivalent to the interdiction on arc $(i_1, i_2)$ in $G'$, which eventually reduces the out flow of $i_2$. For any other arc $(i_2,j_1)$, for any $(i,j)\in A$, in $A'$, the arc-capacity constraints in (2d) become $\sum _{k\in K} {z'}_{ij}^k\le {u'}_{ij}(1-{y'}_{ij})$. Thus, the arc interdiction on arc (i, j) in G is equivalent to the interdiction on arc $(i_2, j_1)$ in $G'$.

Therefore, we have proved the equivalence between model (2) for G with interdiction on both nodes and arcs and model (2) for $G'$ with interdiction only on arcs. $\square$

Appendix 2: Proof of Theorem 1

Proof

Formulation (2) can be formulated equivalently as follows:

$$\begin{aligned} \max _{\mathbf x, \mathbf y;\alpha ,\beta , \gamma }~&\sum _{k\in K}\left( \sum _{i\in S(k)}s_i^k \alpha _{ik}-\sum _{i\in D(k)}d_i^k \alpha _{ik})+\sum _{i\in N}v_i(1-x_i) \beta _i +\sum _{(i,j) \in A}u_{ij}(1-y_{ij}\right) \gamma _{ij} \end{aligned}$$

(13a)

$$\begin{aligned} s.t. ~&\alpha _{ik}-\alpha _{jk}+\beta _i+\gamma _{ij}\le c_{ij}, \quad \forall k\in K, ~(i,j)\in A \end{aligned}$$

(13b)

$$\begin{aligned}&\beta _i , \gamma _{ij} \le 0, \quad \forall i\in N, (i,j)\in A \end{aligned}$$

(13c)

$$\begin{aligned}&\sum _{i\in N}p_{i}x_i+\sum _{(i,j)\in A} q_{ij}y_{ij}\le R \end{aligned}$$

(13d)

$$\begin{aligned}&0\le x_i,y_{ij}\le 1, \quad \forall i\in N, (i,j)\in A \end{aligned}$$

(13e)

By the assumption of R such that all demands can be satisfied for any interdiction plans in ${\mathscr {I}}_c$, the inner problem is always feasible. Through strong duality for the multicommodity flow in the resulted network, the above formulation (13) under this assumption is equivalent to formulation (2) with ${\mathscr {I}}={\mathscr {I}}_c$ for continuous interdiction.

Assume that the set ${\mathscr {I}}_c$ has L extreme points, in the set $ext({\mathscr {I}}_c)=\{(x^{(1)},y^{(1)}),\ldots ,$ $(x^{(L)},y^{(L)})\}$. As the constraints of formulation (13) can be divided into two parts: constraints (13b) and (13c) are only related to $\alpha ,\beta ,\gamma$ while constraints (13d) and (13e) (i.e., ${\mathscr {I}}_c$) are only related to $\mathbf x, \mathbf y$, the objective function (13a) can be reformulated as

$$\begin{aligned} \max _{\alpha ,\beta ,\gamma }\max _{\mathbf x, \mathbf y}\sum _{k\in K}\left( \sum _{i\in S(k)}s_i^k \alpha _{ik}-\sum _{i\in D(k)}d_i^k \alpha _{ik}\right) +\sum _{i\in N}v_i(1-x_i) \beta _i +\sum _{(i,j) \in A}u_{ij}(1-y_{ij})\gamma _{ij} \end{aligned}$$

(14)

which is a linear program for any fixed $\alpha ,\beta ,\gamma$, and the corresponding optimal $\mathbf x, \mathbf y$ should be obtained at some extreme point of ${\mathscr {I}}_c$, and vice versa. Therefore, (14) can be equivalently rewritten by considering constraints in (13) as follows:

$$\begin{aligned} \max _{\alpha ,\beta ,\gamma ;(\mathbf x, \mathbf y)\in ext({\mathscr {I}}_c)} ~&\sum _{k\in K}\left( \sum _{i\in S(k)}s_i^k \alpha _{ik}-\sum _{i\in D(k)}d_i^k \alpha _{ik}\right) +\sum _{i\in N}v_i(1-x_i) \beta _i \nonumber \\&+\sum _{(i,j) \in A}u_{ij}(1-y_{ij})\gamma _{ij} \end{aligned}$$

(15a)

$$\begin{aligned} s.t. ~&\alpha _{ik}-\alpha _{jk}+\beta _i+\gamma _{ij}\le c_{ij}, \quad \forall k\in K, ~(i,j)\in A \end{aligned}$$

(15b)

$$\begin{aligned}&\beta _i , \gamma _{ij} \le 0, \quad \forall i\in N, (i,j)\in A \end{aligned}$$

(15c)

As explained in [24] based on the Dantzig’s theorem in [38], the following conclusion is used for finding solutions of $\mathbf x, \mathbf y$:

For each extreme point $(\mathbf x^{(l)},\mathbf y^{(l)})$ ($l=1,\ldots ,L$) of the polyhedral set ${\mathscr {I}}_c$, there exists a single basic variable in the vector $(\mathbf x^{(l)},\mathbf y^{(l)})$ taking value in the interval [0, 1], while all other variables of $(\mathbf x^{(l)},\mathbf y^{(l)})$ are nonbasic variables taking values of either 0 or 1.

Therefore, the continuous interdiction model is corresponding to the model where only one node or arc is partially interdicted, the rest interdictions are binary. Let $\delta _i,\sigma _{ij}\in \{0,1\}$ associated with $x_i,y_{ij}$ denote whether $x_i,y_{ij}$ take 1 if $\delta _i=1,\sigma _{ij}=1$, or $x_i,y_{ij}$ take 0 if $\delta _i=0,\sigma _{ij}=0$, respectively. Let $\delta '_i,\sigma '_{ij}\in \{0,1\}$ associated with $x_i,y_{ij}$ denote whether $x_i,y_{ij}$ take binary values if $\delta '_i=0,\sigma '_{ij}=0$ or $x_i,y_{ij}$ take any value in [0, 1] if $\delta '_i=1,\sigma '_{ij}=1$, respectively. By the conclusion above, the choices of $\delta '_i$’s and $\sigma '_{ij}$’s should satisfy $\sum _{i\in N}\delta '_i+\sum _{(i,j)\in A}\sigma '_{ij}=1$. Therefore, decision variables $\mathbf x, \mathbf y$ can be replaced by $\delta ,\delta ',\sigma ,\sigma '$ (vectors formed correspondingly), and we have the following equivalent formulation for continuous interdiction through the reformulation of (15) as follows:

$$\begin{aligned} \max _{\delta ,\delta ',\sigma ,\sigma '} ~&\min _{\mathbf z} \sum _{(i,j)\in A} c_{ij} \sum _{k \in K} z_{ij}^k \end{aligned}$$

(16a)

$$\begin{aligned} s.t.~&\text {constraints in (2b)} \end{aligned}$$

(16b)

$$\begin{aligned}&{\left\{ \begin{array}{ll} \sum _{j:(i,j)\in A} \sum _{k} z_{ij}^k / v_i \le 1 - \delta _i - \delta _i^\prime R'/p_i,\quad \forall i\in N\\ \sum _{k\in K} z_{ij}^k/u_{ij} \le 1 - \sigma _{ij} - \sigma _{ij}^\prime R'/q_{ij},\quad \forall (i,j) \in A \\ \end{array}\right. } \end{aligned}$$

(16c)

$$\begin{aligned}&{\left\{ \begin{array}{ll} R'/p_i \le 1 +M_i(1-\delta '_{i}),\quad \forall i \in N\\ R'/q_{ij} \le 1+M_{ij} (1-\sigma '_{ij}),\quad \forall (i,j) \in A\\ \end{array}\right. } \end{aligned}$$

(16d)

$$\begin{aligned}&\sum _{i\in N}p_i \delta _i +\sum _{(i,j)\in A}q_{ij}\sigma _{ij}+R'= R \end{aligned}$$

(16e)

$$\begin{aligned}&\delta _i+\delta '_i \le 1,\quad \forall i \in N \end{aligned}$$

(16f)

$$\begin{aligned}&\sigma _{ij}+\sigma '_{ij} \le 1,\quad \forall (i,j)\in A \end{aligned}$$

(16g)

$$\begin{aligned}&\sum _{i\in N} \delta '_i +\sum _{(i,j)\in A}\sigma '_{ij}=1 \end{aligned}$$

(16h)

$$\begin{aligned}&\delta _i,\sigma _{ij},\delta '_i,\sigma '_{ij} \in \{0,1\},\quad \forall i\in N,(i,j)\in A \end{aligned}$$

(16i)

$$\begin{aligned}&R'\ge 0;~z_{ij}^k \ge 0,\quad \forall (i,j)\in A,\;k\in K \end{aligned}$$

(16j)

The interdiction constraint (16e) limits the budget for all binary interdictions with constraint $R'\ge 0$, where $R'=R-(\sum _{i\in N}p_i \delta _i +\sum _{(i,j)\in A}q_{ij}\sigma _{ij})$ is the remaining budget for a partial disruption of a node or an arc. Constraints (16f) and (16g) indicate that a node and an arc can be either binary interdicted or partially interdicted or not interdicted, respectively. Constraint (16h) limits that only one node or one arc is partially interdicted. The first set of constraints in (16c) include cases for node interdiction: (i) $\delta _i=1,\delta '_i=0$, node i is completely interdicted; (ii) $\delta _i=0,\delta '_i=0$, node i is not interdicted, and this set of constraints limits the node capacity; (iii) $\delta _i=0,\delta '_i=1$, node i is partially interdicted, and this set of constraints limits the capacity after partially interdicted. Similarly, the second set of constraints in (16c) are for arc interdiction. The parameters $M_i$ and $M_{ij}$ in (16d) are relatively large positive constants. The first set of constraints in (16d) ensures that the budget spent on partially interdicted node should be limited by corresponding node interdiction cost, while second set limits the amount for budget spent on partially interdicted arc. The objective function and all other constraints have the same meanings as those in formulation (2).

Let $\alpha _{ik}, \beta _i,\gamma _{ij}$ be the dual variables associated with constraints in (16b) and (16c), under certain R, the formulation (16) is equivalent to :

$$\begin{aligned} \max _{\delta ,\delta ', \sigma ,\sigma ',\alpha ,\beta ,\gamma } ~&\sum _{k\in K}\left( \sum _{i\in O(k)}s_i^k \alpha _{ik}-\sum _{i\in D(k)}d_i^k \alpha _{ik}\right) \nonumber \\&~~+\sum _{i\in N} \beta _{i} \big (1 - \delta _i - \delta _i^\prime R'/p_i \big )+\sum _{(i,j)\in A}\gamma _{ij}\big (1 - \sigma _{ij} - \sigma _{ij}^\prime R'/q_{ij} \big ) \end{aligned}$$

(17a)

$$\begin{aligned} s.t.~&\alpha _{ik}-\alpha _{jk}+\beta _{i}/v_i+\gamma _{ij}/u_{ij} \le c_{ij},\quad \forall (i,j)\in A,\;\forall k\in K \end{aligned}$$

(17b)

$$\begin{aligned}&\text {constraints in (16d)} - \text {(16i)} \end{aligned}$$

(17c)

$$\begin{aligned}&R'\ge 0;~\beta _{i},\gamma _{ij} \le 0,\quad \forall (i,j)\in A,\;\forall k\in K \end{aligned}$$

(17d)

This completes the proof. $\square$