Robust maximum flow network interdiction considering uncertainties in arc capacity and resource consumption

Abstract

This article discusses a robust network interdiction problem considering uncertainties in arc capacities and resource consumption. The problem involves two players: an adversary seeking to maximize the flow of a commodity through the network and an interdictor whose objective is to minimize this flow. The interdictor plays first and selects network arcs to interdict, subject to a resource constraint. The problem is formulated as a bilevel problem, and an upper bound single level mix-integer linear formulation is derived. The upper bound formulation is solved using three heuristics tailored for this problem and the network structure, based on Lagrangian relaxation and Benders’ decomposition. On average, each heuristic provides a reduction in run time of at least 85% compared to a state-of-the-art solver. Enhanced Benders’ decomposition achieves a solution with an optimality gap of less than 5% for all tested instances. Sensitivity analyses are conducted for the level of uncertainty in network parameters and the uncertainty budget. Robust decisions are also compared to decisions not accounting for uncertainty to evaluate the value of robustness, showing a reduction in simulation maximum flows by as much as 89.5%.

Appendices

Appendix A Lagrangian relaxation sub-problems

The sub-problems of the Lagrangian Dual (presented in Sect. 4.2) and their solutions are:

Sub-problem 1:

$$\begin{aligned}&\min _{\alpha ,\beta ,\delta } \quad \left( \sum _{(i,j) \in A'} ({u}_{ij} + \lambda ''_{ij}\hat{u}_{ij})\beta _{ij} + (\lambda 'r_{ij}+\lambda '''_{ij}\hat{r}_{ij})\delta _{ij} \right) - \lambda '\Delta \\&\quad \alpha _i - \alpha _j + \delta _{ij} + \beta _{ij} \ge 0 \quad \forall \,\,\, (i,j) \in A \\&\quad \alpha _t - \alpha _s + \delta _{ts} + \beta _{ts} \ge 1 \\&\quad \alpha _s = 0, \,\, \alpha _t = 1, \,\,\delta _{ts}=0,\,\,\beta _{ts}=0 \\&\quad \alpha _i \in \{0,1\} \quad \forall \,\,\, i \in N \\&\quad \delta _{ij}, \beta _{ij} \in \{0,1\} \quad \forall \,\,\, (i,j) \in A' \end{aligned}$$

The dual of the sub-problem 1 results in a maximum flow problem with updated capacities, as in Bingol (2001).

$$\begin{aligned} SP_1&= \max _{x} \quad x_{ts} - \lambda '\Delta \\&\quad \sum _{(i,j)\in FS_i} x_{ij} - \sum _{(j,i)\in RS_i} x_{ji} = 0 \quad \forall \,\,\, i \in N \\ x_{ij}&\le \min ({u}_{ij}+\lambda ''_{ij}\hat{u}_{ij}\,,\,\lambda 'r_{ij}+\lambda '''_{ij}\hat{r}_{ij}) \quad \forall \,\, (i,j) \in A'\\ x_{ij}&\ge 0 \quad \forall \,\, (i,j) \in A' \end{aligned}$$

Sub-problem 2:

$$\begin{aligned} SP_2&= \min _{\mu } \quad \left( \sum _{(i,j) \in A'}(1-\lambda ''_{ij})\mu _{ij}\right) \\ \mu _{ij}&\in [0,\hat{u}_{ij}] \quad \forall \,\,\, (i,j) \in A' \end{aligned}$$

The solution to the above problem is trivial.

$$\begin{aligned} \mu _{ij}&= {\left\{ \begin{array}{ll} \hat{u}_{ij} &{} ; \text {if}\, \lambda ''_{ij} \ge 1\\ 0 &{} \text {; otherwise} \end{array}\right. } \,\,\, \forall \,\, (i,j) \in A' \end{aligned}$$

Sub-problem 3:

$$\begin{aligned} SP_3&= \min _{\theta } \quad \left( \Gamma -\sum _{(i,j)\in A'}\lambda ''_{ij}\right) \theta \\ \theta&\in [0,\theta _U] \end{aligned}$$

The solution to the above problem is trivial.

$$\begin{aligned} \theta&= {\left\{ \begin{array}{ll} \theta _U &{} ; \text {if}\, \sum \limits _{(i,j) \in A'} \lambda ''_{ij} \ge \Gamma \\ 0 &{} ; \text {otherwise} \end{array}\right. } \end{aligned}$$

Sub-problem 4:

$$\begin{aligned} SP_4&= \min _{\sigma } \quad \left( \sum _{(i,j) \in A'}(\lambda '-\lambda '''_{ij})\sigma _{ij}\right) \\ \sigma _{ij}&\in [0,\hat{r}_{ij}] \quad \forall \,\,\, (i,j) \in A' \end{aligned}$$

The solution to the above problem is trivial.

$$\begin{aligned} \sigma _{ij}&= {\left\{ \begin{array}{ll} \hat{r}_{ij} &{} ; \text {if}\, \lambda '''_{ij} \ge \lambda ' \\ 0 &{} ; \text {otherwise} \end{array}\right. } \,\,\, \forall \,\, (i,j) \in A' \end{aligned}$$

Sub-problem 5:

$$\begin{aligned} SP_5&= \min _{\zeta } \quad \left( \Pi \lambda '-\sum _{(i,j)\in A'}\lambda '''_{ij}\right) \zeta \\ \zeta&\in [0,\zeta _U] \end{aligned}$$

The solution to the above problem is trivial.

$$\begin{aligned} \zeta&= {\left\{ \begin{array}{ll} \zeta _U &{} ; \text {if}\, \sum \limits _{(i,j) \in A'} \lambda '''_{ij} \ge \Pi \lambda ' \\ 0 &{} ; \text {otherwise} \end{array}\right. } \end{aligned}$$

Note that without the upper bounds found in Sect. 4.1, the sub-problems 2–5 could result in unbounded solutions.

Appendix B Simultaneous penalty heuristic for Benders’ decomposition master problem

A heuristic is designed for quick computation of the master problem of Benders’ decomposition. It should be noted that the heuristic solution provides and approximate solution for the master problem. The equivalent formulation of the master problem of Benders’ decomposition as noted in Eqs. 81–89, is given as:

$$\begin{aligned}&\text {Master Problem:} \quad \quad \min _{z,\alpha ,\beta ,\delta } z \end{aligned}$$

(B1)

$$\begin{aligned}&\left( \sum _{(i,j) \in A'} (u_{ij} + \hat{u}_{ij}\gamma ^o_{ij})\beta _{ij} \right) - z \le 0 \quad \forall \,\,\, o \in O \end{aligned}$$

(B2)

$$\begin{aligned}&\left( \sum _{(i,j) \in A'} r_{ij}\delta _{ij} \right) + \left( \sum _{(i,j) \in S} \hat{r}_{ij}\delta _{ij} \right) \le \Delta \quad \forall \,\,\, \{ S \, |\, S \subseteq A',|S |= \Pi \} \end{aligned}$$

(B3)

$$\begin{aligned}&\alpha _i - \alpha _j + \delta _{ij} + \beta _{ij} \ge 0 \quad \forall \,\,\, (i,j) \in A \end{aligned}$$

(B4)

$$\begin{aligned}&\alpha _s = 0, \,\, \alpha _t = 1, \,\,\delta _{ts}=0,\,\,\beta _{ts}=0 \end{aligned}$$

(B5)

$$\begin{aligned}&z \in (-\infty ,\infty ) \end{aligned}$$

(B6)

$$\begin{aligned}&\alpha _i \in \{0,1\} \quad \forall \,\, i \in N \end{aligned}$$

(B7)

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

(B8)

where, O is the set of Benders’ decomposition optimality cuts, and $\gamma ^o_{ij}$ represents $\gamma _{ij}$ values for the cut o. Equation B3 is the resource robustness constraint for the RNIP, written as a set of combinatorially many constraints instead of a single constraint with maximization, which was dualized in order to obtain the final RNIP model formulation (refer MODEL 2 in Sect. 3.4).

The motivation for the heuristic is to account for $|O |$ constraints (in Eq. B2) using a single constraint. This would reduce the master problem to a network interdiction problem with resource consumption uncertainty only (NIPRCU). NIPRCU can be solved by first finding the min-cut of the network using representative capacities, and determining the arc set available for interdiction ($A_I$). Then, determining $\delta$ (the interdiction decision) variables is a robust knapsack problem with arc capacities as item values, resource consumption as item weights, and $\Delta$ is the capacity of the knapsack (solution procedure proposed by Lee et al. (2012) is used for solving the robust knapsack problem). Once, $\delta$ variables are determined, the $\beta$ variables can be easily determined, as they are forward arcs of the minimum cut which are not interdicted.

Calculating representative arc capacities of the network with respect to ${o^{th}}$ optimality cut:

The method simultaneously penalizes the presence of optimality cuts in the set $O\backslash \{o\}$ with respect to the $o^{th}$ optimality cut to calculate the representative arc capacities for the network. The penalty with respect to the $o^{th}$ optimality cut is calculated as:

$$\begin{aligned} cap\_penalty^o = {\left\{ \begin{array}{ll} \displaystyle {\frac{\sum \limits _{(i,j) \in A'}\gamma ^o_{ij}\hat{u}_{ij}}{\sum \limits _{(i,j) \in A'}\gamma ^o_{ij}}} &{} \text {; if } \sum \limits _{(i,j) \in A'}\gamma ^o_{ij} > 0 \\ 0&{} \text {; otherwise} \end{array}\right. } \end{aligned}$$

(B9)

The representative arc capacities of the network with respect to the $o^{th}$ optimality cut ($\tilde{u}^o$) are then given as:

$$\begin{aligned} \tilde{u}^o_{ij} = {\left\{ \begin{array}{ll} u_{ij} + \hat{u}_{ij} &{} \text {; if } \gamma ^o_{ij} = 1 \\ u_{ij} &{} \text {; if } \gamma ^o_{ij} = 0 \text { and } \sum \limits _{q \in O\backslash \{o\}}\gamma ^q_{ij} = 0 \\ u_{ij} + \max \{0, \hat{u}_{ij} - cap\_penalty^o\} &{} \text {; if } \gamma ^o_{ij} = 0 \text { and }\sum \limits _{q \in O\backslash \{o\}}\gamma ^q_{ij} > 0 \\ \end{array}\right. } \end{aligned}$$

(B10)

Steps for solving the Benders’ Master Problem using the simultaneous penalty heuristic:

For each $o \in O$,

• 1. Calculate the representative arc capacities for the network ($\tilde{u}^o$) with respect to optimality cut o.

• 2. Determine the min-cut ($N_s,N_t$) of the network. The variable $\alpha ^o$ and the interdiction arc set ($A_I$) for the network are given as:

  $$\begin{aligned} \alpha ^o_i&= {\left\{ \begin{array}{ll} 1 &{} ; \text {if}\, i \in N_t \\ 0 &{} ; \text {if}\,i \in N_s \end{array}\right. } \\ A_I&= \big \{(i,j) \in A' \,\, \big |\,\, \alpha ^o_j-\alpha ^o_i = 1 \text { and } r_{ij}\le \Delta \text { and } \gamma ^o_{ij}=0 \big \} \end{aligned}$$

• 3. Initialize $\delta ^o$ variables by solving the robust knapsack problem with parameters $c=\tilde{u}^o$, $w = r$, $\hat{w} = \hat{r}$, $b = \Delta$, $N = A_I$, and $\Gamma = \Pi$ (refer Bertsimas and Sim 2003)

• 4. The value of $\zeta ^o$ is the $\Pi ^{th}$ largest value of $\hat{r}_{ij}\delta ^o_{ij}$.

• 5. The values of variables $\sigma ^o$ are given as: $\sigma ^o_{ij}= \max \{0, \hat{r}_{ij} - \zeta ^o \}$

• 6. Determine the $\beta ^o$ variables as follows:

  $$\begin{aligned} \beta ^o_{ij} = {\left\{ \begin{array}{ll} 1 &{} ; \text {if}\, \alpha ^o_j - \alpha ^o_i = 1{ and}\delta ^o_{ij} = 1\\ 0 &{} ; \text {otherwise} \end{array}\right. } \end{aligned}$$

  (B11)

• 7. The value of $z^o$ is $\sum \limits _{(i,j)\in A'}\tilde{u}^o_{ij}\beta ^o_{ij}$

The solution of the master problem is then given as:

$$\begin{aligned} z^*&= \max (z^o) \\ \alpha ^*&= \{\alpha ^o \, |\, z^o = z^* \} \\ \beta ^*&= \{\beta ^o \, |\, z^o = z^* \} \\ \delta ^*&= \{\delta ^o \, |\, z^o = z^* \} \\ \zeta ^*&= \{\zeta ^o \, |\, z^o = z^* \} \\ \sigma ^*&= \{\sigma ^o \, |\, z^o = z^* \} \\ \end{aligned}$$

Appendix C Monte-Carlo simulation for evaluating value of robustness

Algorithm 1

Monte Carlo simulation scheme