Operation Loop-Based Optimization Model for Resource Allocation to Military Countermeasures versus Probabilistic Threat

Chunqi Wan, Xiaoxiong Zhang, Qingsong Zhao, Kewei Yang
2018 Applied Sciences  
Weapons development planning is an unstructured and complex multi-criteria decisionmaking problem, especially in antagonistic environments. In this paper, the defender's decision was modelled as a high complexity non-linear optimization problem with limited resources. An operation loop with realistic link rules was first proposed to model the cooperation relationships among weapons in the defense system. The system dynamics principle was used to characterize the dynamic behavior of the nodes in
more » ... a complex weapons network. Then, we used cumulative threat and development risk to measure different planning solutions by considering the opponent and uncertainties in the development process. Next, an improved Differential Evolution (DE) and Non-Dominated Sorting Differential Evolution (NSDE) were designed to determine the optimal planning solutions for a single objective and multi-objective. The compromise solution, based on the Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS), was used to evaluate the Pareto solution set of the multi-objective. Finally, an illustrative case was studied to verify the feasibility and validity of the proposed model. race between two parties, R and B. Each party considers the potential threat from the other party. R develops one or more weapons to gain operational advantages over B, whereas B attempts to develop one or more CMs that would neutralize, or at least mitigate, the threat from R [7]. Resource allocation in adversarial settings, also called the research of arms races using mathematical models, has been studied since the 1930s. However, most of these studies focused only on the strategic aspect but not the operational aspect. In 1935, Richardson [8] first proposed a simple coupled differential equation model to solve the effects of an arms race between rival states, thus successfully predicting World War II. Later, Hunter [9] analyzed a three-nation arms race by solving linear difference equations based on Richardson's study. Similar to most previous research, Hunter's model focused only on the strategic aspect and did not consider the relationship between research and development (R&D) investments and time, nor the cooperation relationships among different weapons. Afterwards, Etcheson [3] established and solved equations to optimize the function of each side rather than the constrained nonlinear optimization model. Recently, Paulson et al. [10] proposed a model for determining optimal resource allocation by combining game theory with a simple multi-attribute utility model. The defender first allocated her resources amongst all combinations of countermeasures and targets, and then the attacker striked with his best response to this allocation. Golany [11] investigated efficient computation schemes for allocating two defensive resources to multiple sites to protect against possible attacks by an adversary. The availability of the two resources was constrained and the effectiveness of each may vary over the site. Mazicioglu [12] modelled the attacker's behavior using multi-attribute prospect theory to account for the attacker's multiple objectives and deviations from rationality. These articles mainly modelled adversary's behavior from macroscopic and strategic view, and considered neither the time required to develop new countermeasures, nor the relationship between expenditures and development gap. In addition, budgets were not modelled explicitly. WSOS is a variety of weapons that can be functionally related to each other in accordance with a certain structure with a higher level of integration with certain strategic guidance, operational command, and security conditions [13] . Moreover, with the development of technology, the operational effectiveness of an army depends on the interactions between multiple weapons instead of independent weapons [14] . However, to the best of our knowledge, the relationships among different weapons have rarely been modelled in the literature. For example, the most authoritative article in Operations Research [15] focused only on a single family of countermeasures (e.g., interception systems or bomb neutralization systems) that evolves and improves over time. With the development of informationization, the performance of defense depends on the interaction of multiple systems rather than the individual attributes of the armaments. As a consequence, the defender tends to use several families of CMs instead of a single type, so the performance of the defense cannot be determined only by the effectiveness of any single CM employed. Instead, the cooperation between CMs should be considered. Furthermore, as new CMs are introduced, even in the decision-making stage, the defender may also have to consider the interaction between the current CMs and future possible CMs to optimize the defense system. If the relationships among weapons are ignored in the development planning process, the problem will be simplified to a simple selective problem of a single weapon, neglecting the holistic structural characteristics [16] . Thus, we used and improved an operation loop-based network model focusing on the cooperation relationships among the weapons [17], in which weapons are represented as nodes, whereas relationships among the weapons are modelled as edges. In this model, weapons entities contain sensor, decision, influencer, and target nodes. All types of nodes are treated as target nodes to the enemy. Note that one weapon may be characterized as multiple nodes, and each node is attached to a parameter, denoting the amount of the resources allocated to it. In addition, we considered the real link rules of these nodes that prior studies [18] did not mention. For example, if more than one S node exists in one operation loop, the nodes could only be arranged based on their index values of Radius in descending order. Due to the limitation of the geographical locations of the different weapons and some other factors, not all nodes could transfer information or energy, Appl. Sci. 2018, 8, 214 3 of 25 and certain nodes could only be arranged in descending or ascending order. This phenomenon is common in real-life operational battles. In summary, this problem can be modelled as a constrained nonlinear optimization problem, which is known as NP-hard. The complexity of this type of problem depends on the complex relationships among weapons. Consequently, we argue that the analytical solution to the problem is not feasible, and a heuristic-based optimization algorithm would be effective. In this article, we used Differential Evolution (DE) and Non-Dominated Sorting based Genetic Algorithm-II (NSGA-II) for our optimization tasks. The main contributions of this paper are as follows: (1) When assessing the threat posed by the enemy, we not only considered the cooperation among different nodes based on the traditional ideology of the operation loop, but also the real link rules of these nodes. Meanwhile, the system dynamics principle was used to characterize the dynamic behavior of the nodes in the complex weapons network; (2) According to the characteristics of the weapon life cycle curve, we considered some realistic constraint conditions, such as the annual weapons budget, the total weapons budget, and the weapon planning cycle; (3) A complex multi-objective planning problem and the related solving approach were proposed, considering the development risks and probabilistic threat from the enemy. In this paper, we considered practical settings in realistic dynamic competition. This paper was organized as follows. Section 2 introduced the modeling method based on operation loop and the threat assessment process. Section 3 elaborated upon the single objective optimization model with time and investment constraints. In addition, the algorithm to solve the problem was outlined. Section 4 extended the objective functions to an actual weapons R&D process. Section 5 demonstrated the usefulness of the models by presenting the results of a numerical experiment. Finally, Section 6 concluded the paper and discussed possible future research.
doi:10.3390/app8020214 fatcat:brkpsgtsfjew7j4v5geldfqdfe