**3. Weapon target assignment (WTA) problem**

The assignment problem is one of the fundamental constrained combinatorial optimization problems in the branch of optimization or operation research in Mathematics. This problem is mainly used in decision making. Here we consider a special type of problem which is a combination of transportation problem and assignment problem. By the name of weapon-target assignment problem, it is clear that we have to assign weapons to targets. It is a defense-related application in operation research and is slightly different from the more general optimal resource allocation problem. The main aim of weapon-target assignment problem is to find a set of solution of the number of available weapons to a set of required targets so that the expected rewards of the sequential engagement is maximized [6]. The engagement of a target by a weapon is modeled as a stochastic event, with a probability of kill assigned to each weapon-target pair (this is the probability that the interceptor weapon will destroy the target if assigned to it). The engagement of a weapontarget pair is independent of all other weapons and targets. This is an integer optimization problem in that fractional weapon assignments are not allowed.

assigned to more than one target. But, when targets are assigned to more than one targets or targets, are assigned to more one weapon, then the assignment problem

Weapon target assignments are generally viewed as nonlinear assignment problems (non-LAP). That is, the optimal solution is nonlinear but is still considered to

A WTA problem can be viewed from either a target-based or an asset-based perspective. In the target-based, values are assigned to each target to cause damage to the defended asset. The objective of the target-based WTA solution is to maxi-

Conversely, in an asset-based perspective values are assigned to the assets rather than the targets. This WTA problem is where weapons are assigned such that the combined value of assets is maximized. The asset-based approach requires information on which targets are approaching the defended assets. But a target-based approach is more appropriate than the asset-based. The approach which is discussed

becomes nonlinear as presented by the bipartite graph in **Figure 2 (b)**.

integer values as in the LAP case.

*A linear and a nonlinear bipartite graph.*

*Weapon Target Assignment*

*DOI: http://dx.doi.org/10.5772/intechopen.93665*

**Figure 2.**

*3.1.2 Asset-based versus Target-based*

mize the damage value of the incoming targets.

in this chapter is the target-based perspective.

Generally, WTA is categorized into two versions:

*3.1.3 Static versus dynamic*

a. Static WTA

**185**

b. Dynamic WTA

#### **3.1 Basic factors of WTA**

A number of different approaches have been applied to the WTA problem. When considering a WTA problem, a number of factors need to be considered. Some of these factors are discussed below [7]:

#### *3.1.1 Linear versus non-linear assignment problem*

The generalized linear assignment problem (LAP) of allocating weapons to targets is a fundamental problem of combinatorial optimization. In the simplest case, the number of weapons and targets are equal, with only one weapon being assigned to any one target in an allocation. LAP's can also be represented in a bipartite graph shown in **Figure 2 (a)**. In the LAP graph, weapons cannot be

models are mainly studied and the dynamic WTA are not fully studied indeed. In the research on algorithms of WTA, the intelligent algorithms are often used to

WTA have been proposed for several years is shown in **Table 1**.

*Concepts, Applications and Emerging Opportunities in Industrial Engineering*

algorithm for the WTA problem based on the kill probabilities.

**3. Weapon target assignment (WTA) problem**

There are so many proposed algorithms on WTA problem [4, 5]. So we present the summary of variant heuristic algorithms and the implementation of various

Various combinatorial optimization techniques are currently available. Most of these techniques have not been thoroughly tested on realistic problems. In this chapter, we consider a class of non-linear assignment problems collectively referred to as Target-based Weapon Target Assignment (WTA). We first briefly discuss the weapon target assignment problem. We also include the basic concepts and models of WTA and the mathematical nature of the WTA models is also analyzed. We present some real-life applications of WTA here. There does not exist any exact methods for the WTA problem even relatively small size problems, and much research has focused on developing heuristic algorithms based on meta-heuristic techniques. The main focus of this chapter is our new developed optimization

The assignment problem is one of the fundamental constrained combinatorial optimization problems in the branch of optimization or operation research in Mathematics. This problem is mainly used in decision making. Here we consider a special type of problem which is a combination of transportation problem and assignment problem. By the name of weapon-target assignment problem, it is clear that we have to assign weapons to targets. It is a defense-related application in operation research and is slightly different from the more general optimal resource allocation problem. The main aim of weapon-target assignment problem is to find a set of solution of the number of available weapons to a set of required targets so that the expected rewards of the sequential engagement is maximized [6]. The engagement of a target by a weapon is modeled as a stochastic event, with a probability of kill assigned to each weapon-target pair (this is the probability that the interceptor weapon will destroy the target if assigned to it). The engagement of a weapontarget pair is independent of all other weapons and targets. This is an integer optimization problem in that fractional weapon assignments are not allowed.

A number of different approaches have been applied to the WTA problem. When considering a WTA problem, a number of factors need to be considered.

The generalized linear assignment problem (LAP) of allocating weapons to targets is a fundamental problem of combinatorial optimization. In the simplest case, the number of weapons and targets are equal, with only one weapon being assigned to any one target in an allocation. LAP's can also be represented in a bipartite graph shown in **Figure 2 (a)**. In the LAP graph, weapons cannot be

solve the WTA problem.

**3.1 Basic factors of WTA**

**184**

Some of these factors are discussed below [7]:

*3.1.1 Linear versus non-linear assignment problem*

**2. Weapon's target assignment**

**Figure 2.** *A linear and a nonlinear bipartite graph.*

assigned to more than one target. But, when targets are assigned to more than one targets or targets, are assigned to more one weapon, then the assignment problem becomes nonlinear as presented by the bipartite graph in **Figure 2 (b)**.

Weapon target assignments are generally viewed as nonlinear assignment problems (non-LAP). That is, the optimal solution is nonlinear but is still considered to integer values as in the LAP case.
