A stochastic bilevel model for facility location-protection problem with the most likely interdiction strategy

Highlights

• Propose a stochastic bilevel model for the facility location and protection problem.

• Allow for the capacitated facility, bounded rationality and multi-state capacity.

• Involving probabilistic fortification and interdiction makes the uncertainty.

• The minimum information approach determines the most likely interdiction strategy.

• Design a hybrid genetic algorithm for dealing with the intractable model.

Abstract

This paper studies a capacitated facility location and protection problem against the boundedly rational attacker that merges probabilistic fortification and interdiction, and decision-dependent uncertainty in the post-interdiction capacity of the facilities. The defender decides the location and fortification of the facilities, whereas the attacker uses the minimum information approach to determine the amount of interdiction exerted for facilities based on the interdiction exertion probability. The defender and attacker’s strategy decisions influence the capacity of facilities, making decision-dependent uncertainty. The problem is formulated as a two-stage stochastic bilevel model. The first stage involves a bilevel game formulation where the defender and attacker make decisions sequentially at the outer and inner-level, respectively. The second stage is to optimize the assignments of the facilities with decision-dependent uncertain capacity state probability. A novel hybrid genetic algorithm is developed to solve the proposed model. Also, an analytic hierarchy process and a multinomial logistic regression are embedded into the algorithm for estimating the probability of interdiction exertion and uncertain post-interdiction capacity state, respectively. Computational experiments are provided to illustrate the validity of the proposed model.

Introduction

Critical infrastructure facilities in the supply networks, such as power grid, biomedical devices, and post offices, are threatened by the rapidly growing terrorism in recent years. After being interdicted, the losses of the supply network function have happened. To reduce the threat of terrorist attacks, there is a growing interest in locating and/or protecting facilities against imminent attacks. The problem is regarded as facility interdiction problems which have been widely modeled by game theory [1], [2], [3], [4], [5]. However, the existing studies usually consider location and protection decisions separately, and most of them provide solutions based on assumptions that fail to address real-life problems. Such assumptions include (1) facilities are uncapacitated and can provide unlimited services and resources to the demand centers (uncapacitated facility), (2) the attacker is a perfectly rational player who prefers conducting the worst case of interdiction against facilities (perfectly rational attacker), (3) facilities are completely protected if fortified and fail completely once exposed to an interdiction (perfect fortification and interdiction), (4) after an interdiction, facilities are either operational with full functionality or non-operational with the losses of function (binary-state capacity). However, in reality, there are a set of constraints on the capacities of facilities that could limit the services of facilities. And the assumption of perfect rationality would provide non-robust strategies for the defender because the attacker has biases that affect his decision-making processes. Further, the fortification does not provide perfect defense against intentional attacks. Similarly, the interdiction also has no ability to disrupt the facility completely. Thereby, the facilities can be partially operational after interdiction, depending on the fortification level and the total exerted interdiction amount. Thus, the existing studies based on these assumptions are likely to provide inferior decisions about the location/fortification of facilities in response to intentional attacks. In this regard, we relax these assumptions and extend the literature by taking the capacitated facility into account, considering the joint location and fortification decisions, incorporating a boundedly rational attacker. Furthermore, probabilistic fortification and interdiction, multiple levels of fortification, and multiple types of interdiction are allowed to make the uncertainty in the post-interdiction capacity of facilities.

In terms of facility interdiction models, location and protection are two mitigation strategies to decrease the impact of intentional disruption risks. Location strategy is implemented in the design phase of the supply networks. O’Hanley and Church [2] were one of the pioneers who formulated a bilevel programming to decide where facilities to locate for hedging against worst-case facility losses.  Meng et al. [6] proposed a mixed integer programming to determine the optimal location of terror response facilities when subject to both unintentional and intentional disruptions. This work was extended by Li et al. [7] where hidden information about facilities is taken into account. Ghaffarinasab and Motallebzadeh [8] investigated three different variants of hub interdiction problem in a multiple allocation hub-and-spoke, which was further extended by Ullmert et al. [9]. Xiang and Wei [10] studied an integrated emergency facility location and network interdiction problem where the attacker is allowed to use an interdicted road segment.

The concept of protection strategy allocates resources across targets to fortify against terrorism threats [11]. Church and Scaparra [12] first introduced the r-interdiction median problem with fortification to optimally allocate defensive resources to mitigate disruption risks stemming from the attack. According to this fortification model, extensive previous efforts such as uncertain number of attacks [13], probabilistic protection [14] and defensive budget constraint [15] were developed. Moreover, multiple allocation hub interdiction problems in hub-and-spoke networks have been conducted [16], [17], [18]. Khanduzi and Maleki [19] offered a multi-period interdiction problem with fortification that involves a dynamic Stackelberg game based on capacity recovery. In each period, the defender provides services to customers and allocates fortification resources to facilities that are exposed to possible interdiction exerted by the attacker. Parajuli et al. [20] formulated a tri-level mixed integer programming model for the protection problem that seeks to secure and backup facilities with appropriate capacity levels and response speeds under disruptions. A thread of literature that concerns on protection strategy at different systems consist of literature [21], [22], [23], [24], [25].

Note that, all of the above-mentioned models deal with the protection and location strategies separately that fail to provide reliable decisions against attacks [26]. Thus, the joint consideration of the facility location and protection strategies is necessary. Mahmoodjanloo et al. [27] and Akbari-Jafarabadi et al. [28] proposed an r-interdiction median problem combining location and protection decisions, in which complete interdiction and probabilistic protection were modeled. Jalali et al. [29] formulated a convex risk-averse location-protection problem that a risk measure is incorporated into the decision-making. Zhang et al. [30] presented a multi-objective optimization for location and protection of facilities simultaneously, where secrecy is used into a simultaneous game so that each player moves without knowing their adversary’s strategies. In these location and/or protection models, the uncapacitated facility has been widely investigated. Although there exist some studies on the capacitated facility [19], [27], [28], [31], most of them assumed that the capacity of facilities could be decreased to zero when exposed to an interdiction, i.e., the interdiction is perfect. Despite Hausken and Levitin [32] have investigated multi-state system survivability theory, each element in the system is still assumed to have two states: perfect functioning and total failure. However, the element may have another state, such as partial operation. Thus, we assume that the capacitated facility can remain partially operable after interdiction, depending on the fortification level and the total exerted interdiction amount. In this regard, we allow probabilistic fortification and interdiction to make uncertainty in the post-interdiction capacities of facilities for improving the robustness of the model.

According to the literature above, a defender–attacker sequential game has been widely applied in the field of the facility interdiction problem. However, all of these game models assume that the attacker is perfectly rational and would always choose a pure strategy to maximize his expected utility or maximize the defender’s loss. In reality, the attacker’s behavior would be influenced by many factors, such as the degree of rationality, risk preference, and intelligence information [33], [34]. Thus, within the defender–attacker game, the behavioral model is worth studying. Nikoofal and Gumüs̈ [35] studied a game-theoretic model by considering information asymmetry and discussed how the attacker’s degree of rationality and target preference affect defense equilibrium. Zhang et al. [36] used a quantal response model to model the attacker’s bounded rationality for multi-target defensive resource allocation in defender–attacker game. Cheung and Bell [37] also used the concept of quantal response to allow for biases when defining attack probabilities in defending critical assets. Zhang et al. [38] studied the risk preferences of both defender and attacker in a sequential game. They analyzed the effect of risk preferences on the players’ behavior in equilibrium. Lei et al. [39] utilized conditional value-at-risk to deal with a generic maximum flow interdiction problem, where the performance of solutions by the leader and the follower is measured in the left-tail and right-tail CVaR values, respectively. The risk measure was also used by Jalali et al. [29] accounting for facility location and protection problem under interdiction. There is little literature applying the behavioral model within the defender–attacker game to facility interdiction problem. In this paper, we introduce a new approach, namely the minimum information (MI) approach inspired by [40], to model the attacker’s decision-making behavior. The prior information is given as the probabilities for attacking each facility by different interdiction types, which may be determined by considering some attributes, including but not limited to the degree of rationality and risk preference, taken into account by multiple attributes decision-making procedures.

As described by Hausken and Levitin [41], systems defense and attack models were methodically classified according to the system structure, defense measures, and attack tactics and circumstances. Theoretically, within the classifications proposed in Hausken and Levitin [41], the system structure, defense measures, and attack tactics and circumstances considered in this paper fall under the category of multiple elements, protection, and attack against multiple elements, respectively. Furthermore, by scrutinizing the theoretical literature, this paper makes analyses similar to the analysis in [32], [42], [43], [44]. In detail, this paper is devoted to studying facility interdiction problem jointly considering location and fortification decisions and involving the capacitated facility, probabilistic fortification and interdiction, as well as bounded rationality. The defender decides the location and fortification of facilities. Whereas the attacker is a boundedly rational player who exerts a certain amount of interdiction for established facilities according to his most likely interdiction strategy. Note that the probabilistic fortification and interdiction make the uncertain post-interdiction capacity of facilities constructing scenario set. To this end, we first propose a two-stage stochastic bilevel programming model (TSSBP) with decision-dependent uncertainty, where the first stage is modeled as a bilevel game formulation that the defender and attacker make decisions sequentially at the outer and inner-level, respectively. The defender-level objective function is devoted to minimizing the total cost of locating and fortifying facilities and serving demand centers. While the attacker level problem is modeled by the MI approach to determine the most likely interdiction strategy. After realizing a specif configuration of location, fortification and interdiction strategies, the post-interdiction capacities of the facilities can be obtained, and a set of scenarios is constructed. Each scenario is constructed by a particular combination of the post-interdiction capacities of the facilities. The total number of scenarios is equal to the number of possible combinations of the facilities’ post-interdiction capacities. A scenario-based minimization problem for the user is presented in the second stage of the model. The minimization problem seeks to minimize the network cost by optimally assigning demand centers to facilities bases on existing information.

The resulting TSSBP model is a nonlinear and NP-hard problem, which is acknowledged as a challenging problem to solve. A similar structure of the model that can be found in the literature is [45], where decision-dependent uncertainty is not involved in comparison to our TSSBP model. Thus, the multi-objective solution method proposed in [45] cannot be used in this paper. In this regard, we develop a novel hybrid genetic algorithm that is used to iteratively solve the defender and attacker level problem in the first stage of the model. In detail, the outer-level GA proceeds to obtain the optimal defense strategies. And the inner-level GA is designed to generate an approximation to the attacker’s most likely interdiction strategy, given a defense strategy. Moreover, an analytic hierarchy process(AHP) and a multinomial logistic regression are embedded into the hybrid GA to estimate the probability of interdiction exertion and post-interdiction capacity state, respectively.

Compared with previous literature, this paper has the following significant contributions: (1) We develop facility interdiction problem jointly considering location and protection decisions under the capacitated facility and bounded rationality for practical implications. (2) Involving probabilistic fortification and interdiction makes the decision-dependent uncertainty in the post-interdiction capacity states of the facilities. The probability of post-interdiction capacity states is estimated by a predictive modeling technique. (3) The most likely interdiction strategy of a boundedly rational attacker is determined by the minimum information approach. The probability of exerting interdiction for facilities is estimated by the analytic hierarchy process. (4) A hybrid genetic algorithm is proposed for dealing with the intractable two-stage stochastic bilevel programming model.

For ease of reference, we provide the necessary notation used in this paper in Table 1.

The remainder of this paper is organized as follows. In Section 2, the MI approach for determining the most likely interdiction strategy, decision-dependent scenario probability as well as mathematical programming formulation are stated. In Section 3, a hybrid genetic algorithm is introduced to solve the TSSBP model. Moreover, the probabilities of interdiction exertion and capacity states are estimated. In Section 4, several computational experiments are provided for the purpose of illustration. In the end, Section 5 summarizes the paper and presents some possible works for future research.