*3.1.3 Static versus dynamic*

Generally, WTA is categorized into two versions:


**Static WTA:** In the static version, all of the inputs (i.e., weapons, targets, desired effects, engagement time, etc.) to the problem are fixed, and all weapons are engaged to targets in a single stage. The damage assessment is made when all the weapons are engaged to the targets completely. Thus the main objective of SWTA [8] is to find the proper assignment of a temporary defense task. That is, in static WTA the optimal assignment of weapons to targets only allowed a single weapon to be assigned to a single target. Then the static one can be considered as a constrained resource assignment problem. The static version of the problem is a special case of the dynamic one.

Symbols: Descriptions.

*Weapon Target Assignment*

*pij*

assigned to target

Here we consider *pij*

target *<sup>j</sup>*, therefore the term 1 � *pij*

the expected damage to target *<sup>j</sup>* is 1 � <sup>Q</sup>*<sup>W</sup>*

objective function and linear constraints.

*3.2.1 Applications of WTA*

in the current section:

**187**

*W*: The number of Weapon types.

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

assignment and used to priorities target engagement.

to maximize the expected damage value of targets.

*T*: The number of targets that must be assigned by weapons.

*tj*: The minimum number of weapons required to target *j*.

gramming formulation in terms of the above-introduced variables,

*j*¼1

*T*

*j*¼1

max X *T*

*subject to*, X

X *W*

*i*¼1

*u <sup>j</sup>*: The military value of the target *j*s. This is determined during the weapon

*kill*: The destroying probability of target *j* by weapon of type*i*, also expressed as the kill probability for weapon type *i* on target *j*. It's given for all weapons and targets. *xij*: An integer decision variable indicating the number of weapons of type *i*

*j*: That is, how many numbers of weapons of type *i* should be assigned to target *j*

Let there be *T* targets numbered as 1, 2, … , *T* and *W* weapon types numbered 1, 2, … ,*W*. Then we can now formulate the objective function in terms of probability of damage of various targets weighted by their military value. So the weapon target assignment may now be modeled as the following nonlinear integer pro-

> *<sup>u</sup> <sup>j</sup>* <sup>1</sup> � <sup>Y</sup> *W*

weapon *i* assigned to it. By the over-all assignment of weapons of all types, P*<sup>W</sup>*

probability of the total expected damaged value of the targets is being represented by the objective function (3). Limitations on the number of weapons assigned are specified in terms of *wi* and *tj*. The constraints represented above by Eqs. (4) and (5) are on weapons available of various types and on the minimum number of weapons to be assigned to various targets. By the Eq. (4), we can assure that the total number of weapons used does not exceed the available capacity, and as well as the Eq. (5) ensures that the total number of weapons should exceed the minimum number of weapons required for target*j*s. Eq. (6) provides the non-negativity of decision variables. Here we observe that the resulting problem has non-linear

The WTA problem has wide applications in real life. Some of them are discussed

*<sup>i</sup>*¼<sup>1</sup> <sup>1</sup> � *pij*

*i*¼1

<sup>1</sup> � *pij*

*xij* ≥0, int*eger*, *i* ¼ 1, … ,*W and j* ¼ 1, … , *T* (6)

*kill* as the destroying probability by weapons of type *i* on

*kill* � � denotes the survival probability for target *<sup>j</sup>* if

*kill* � �*xij* h i. The maximization of the

*<sup>i</sup>*¼<sup>1</sup>*xij*

*kill* � �*xij* " #, (3)

*xij* ≤ *wi i* ¼ 1, *:* … , *W* (4)

*xij* ≥*tj j* ¼ 1, *:* … , *T* (5)

*wi* : The number of weapons of type *i* available to be assigned to targets.

**Dynamic WTA:** Dynamic WTA problem is originally proposed by Hosein and Athans in 1990 [9], and attract much more attention from researchers in recent years. The goal of DWTA is to find a global optimal assignment for the whole defense process in which the engagement occasion of weapons must be taken into account. The dynamic WTA can also be expressed as a succession of static WTA. That is, in dynamic WTA there are no restrictions as discussed before in SWTA problem. Many weapons can be assigned to a single target. This satisfies the real scenario of a defense system. When the scale is large, the DWTA models are comparatively more complex than the SWTA models. In this paper, we mainly focus on the dynamic weapon-target assignment problem.

In addition, considering the different missions, each version includes the assetbased problem and the target-based problem. In the asset-based problem, the task is to maximize the expected total value of assets which are defended by the defensive weapons. In the target-based problem, the task is to minimize the expected total value of targets which are not destroyed by the defensive weapons after the engagement. The target-based problem can be considered as a special case of the asset-based problem.

### *3.1.4 Properties of dynamic WTA*

Some relevant properties of the dynamic WTA problem are that it is:


These properties of the problem rule out any hope of obtaining efficient optimal algorithms.

#### **3.2 Mathematical formulation of WTA**

To present the dynamic weapon-target assignment problem, we need the following parameters and variables to be introduced:

Symbols: Descriptions.

**Static WTA:** In the static version, all of the inputs (i.e., weapons, targets, desired effects, engagement time, etc.) to the problem are fixed, and all weapons are engaged to targets in a single stage. The damage assessment is made when all the weapons are engaged to the targets completely. Thus the main objective of SWTA [8] is to find the proper assignment of a temporary defense task. That is, in static WTA the optimal assignment of weapons to targets only allowed a single weapon to be assigned to a single target. Then the static one can be considered as a constrained resource assignment problem. The static version of the problem is a special case of

*Concepts, Applications and Emerging Opportunities in Industrial Engineering*

**Dynamic WTA:** Dynamic WTA problem is originally proposed by Hosein and Athans in 1990 [9], and attract much more attention from researchers in recent years. The goal of DWTA is to find a global optimal assignment for the whole defense process in which the engagement occasion of weapons must be taken into account. The dynamic WTA can also be expressed as a succession of static WTA. That is, in dynamic WTA there are no restrictions as discussed before in SWTA problem. Many weapons can be assigned to a single target. This satisfies the real scenario of a defense system. When the scale is large, the DWTA models are comparatively more complex than the SWTA models. In this paper, we mainly

In addition, considering the different missions, each version includes the assetbased problem and the target-based problem. In the asset-based problem, the task is to maximize the expected total value of assets which are defended by the defensive weapons. In the target-based problem, the task is to minimize the expected total value of targets which are not destroyed by the defensive weapons after the engagement. The target-based problem can be considered as a special case of the

Some relevant properties of the dynamic WTA problem are that it is:

resort to complete enumeration to find the optimal solution.

b. Discrete (fractional weapons assignment are not allowed)

d. Nonlinear (the objective function is convex)

enumeration techniques impractical).

lowing parameters and variables to be introduced:

**3.2 Mathematical formulation of WTA**

a. NP-Complete (Non-deterministic polynomial), that is one must essentially

c. Dynamic (the results of previous engagements are observed before making

e. Stochastic (weapon-target engagements are modeled as stochastic events)

These properties of the problem rule out any hope of obtaining efficient optimal

To present the dynamic weapon-target assignment problem, we need the fol-

f. Large-Scale (the number of weapons and targets is large, making

focus on the dynamic weapon-target assignment problem.

the dynamic one.

asset-based problem.

*3.1.4 Properties of dynamic WTA*

present assignments)

algorithms.

**186**

*W*: The number of Weapon types.

*T*: The number of targets that must be assigned by weapons.

*u <sup>j</sup>*: The military value of the target *j*s. This is determined during the weapon assignment and used to priorities target engagement.

*wi* : The number of weapons of type *i* available to be assigned to targets.

*tj*: The minimum number of weapons required to target *j*.

*pij kill*: The destroying probability of target *j* by weapon of type*i*, also expressed as the kill probability for weapon type *i* on target *j*. It's given for all weapons and targets.

*xij*: An integer decision variable indicating the number of weapons of type *i* assigned to target

*j*: That is, how many numbers of weapons of type *i* should be assigned to target *j* to maximize the expected damage value of targets.

Let there be *T* targets numbered as 1, 2, … , *T* and *W* weapon types numbered 1, 2, … ,*W*. Then we can now formulate the objective function in terms of probability of damage of various targets weighted by their military value. So the weapon target assignment may now be modeled as the following nonlinear integer programming formulation in terms of the above-introduced variables,

$$\max \quad \sum\_{j=1}^{T} u\_j \left[ 1 - \prod\_{i=1}^{W} \left( 1 - p^{ij}\_{kill} \right)^{x\_{ij}} \right], \tag{3}$$

$$\text{subject to}, \qquad \sum\_{j=1}^{T} \mathbf{x}\_{ij} \le w\_i \qquad i = 1, \ldots, W \tag{4}$$

$$\sum\_{i=1}^{W} \mathbf{x}\_{ij} \ge \mathbf{t}\_j \qquad j = \mathbf{1}, \dots, T \tag{5}$$

*xij* ≥0, int*eger*, *i* ¼ 1, … , *W and j* ¼ 1, … , *T* (6)

Here we consider *pij kill* as the destroying probability by weapons of type *i* on target *<sup>j</sup>*, therefore the term 1 � *<sup>p</sup>ij kill* � � denotes the survival probability for target *<sup>j</sup>* if weapon *i* assigned to it. By the over-all assignment of weapons of all types, P*<sup>W</sup> <sup>i</sup>*¼<sup>1</sup>*xij* the expected damage to target *<sup>j</sup>* is 1 � <sup>Q</sup>*<sup>W</sup> <sup>i</sup>*¼<sup>1</sup> <sup>1</sup> � *pij kill* � �*xij* h i. The maximization of the probability of the total expected damaged value of the targets is being represented by the objective function (3). Limitations on the number of weapons assigned are specified in terms of *wi* and *tj*. The constraints represented above by Eqs. (4) and (5) are on weapons available of various types and on the minimum number of weapons to be assigned to various targets. By the Eq. (4), we can assure that the total number of weapons used does not exceed the available capacity, and as well as the Eq. (5) ensures that the total number of weapons should exceed the minimum number of weapons required for target*j*s. Eq. (6) provides the non-negativity of decision variables. Here we observe that the resulting problem has non-linear objective function and linear constraints.
