**3.6 WTA on real battle field**

On modern battlefields, the task of battle managers is very important to make a proper assignment of weapons to targets to defend own-force assets or to offend the opponent targets. As an example, we now consider a target-based weapon-target assignment model for maximizing the total expected damage value of the targets which satisfies the Eqs. (3)–(5). Here considering five weapons are to be assigned to 20 targets [17, 18]. These targets have different probabilities of killing to platforms which are dependent on the target types. That is, the destroying probabilities of targets by different types of weapons obviously will be different. The probabilities define the effectiveness of the *ith* weapon to *jth* destroy the target. Here we get by the weapon-target pair that there are total 100 variables that are to be found out. The upper limits on weapon capacity and lower limits on weapons to be assigned are also given.

The characteristics of the five weapon types could be thought as follows:

1.Breda-SAFAT machine gun

2.Lewis gun

3. Spandau machine gun


Each weapon-target pair survival probabilities are shown in **Figure 3**.

**3.7 Formulation of battle field example**

*Minimum requirements of weapons assigned to targets.*

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

age value as follows:

*Weapon Target Assignment*

**Figure 5.**

subject to,

**193**

After having all the values of required parameters, we formulate the model corresponding to the given example for maximizing the total expected target dam-

> 150 1*:*<sup>00</sup> � <sup>1</sup>*<sup>x</sup>*1,10 � <sup>0</sup>*:*86*<sup>x</sup>*2,10 � <sup>0</sup>*:*95*<sup>x</sup>*3,10 � <sup>0</sup>*:*96*<sup>x</sup>*4,10 � <sup>0</sup>*:*90*<sup>x</sup>*5,10 <sup>½</sup> ð Þ�þ 30 1*:*<sup>00</sup> � <sup>1</sup>*<sup>x</sup>*1,11 � <sup>1</sup>*<sup>x</sup>*2,11 � <sup>0</sup>*:*99*<sup>x</sup>*3,11 � <sup>0</sup>*:*91*<sup>x</sup>*4,11 � <sup>0</sup>*:*95*<sup>x</sup>*5,11 <sup>½</sup> ð Þ�þ 45 1*:*<sup>00</sup> � <sup>1</sup>*<sup>x</sup>*1,12 � <sup>0</sup>*:*98*<sup>x</sup>*2,12 � <sup>0</sup>*:*98*<sup>x</sup>*3,12 � <sup>0</sup>*:*92*<sup>x</sup>*4,12 � <sup>0</sup>*:*96*<sup>x</sup>*5,12 <sup>½</sup> ð Þ�þ 125 1*:*<sup>00</sup> � <sup>1</sup>*<sup>x</sup>*1,13 � <sup>1</sup>*<sup>x</sup>*2,13 � <sup>0</sup>*:*99*<sup>x</sup>*3,13 � <sup>0</sup>*:*91*<sup>x</sup>*4,13 � <sup>0</sup>*:*91*<sup>x</sup>*5,13 <sup>½</sup> ð Þ�þ 200 1*:*<sup>00</sup> � <sup>1</sup>*<sup>x</sup>*1,14 � <sup>0</sup>*:*88*<sup>x</sup>*2,14 � <sup>0</sup>*:*98*<sup>x</sup>*3,14 � <sup>0</sup>*:*92*<sup>x</sup>*4,14 � <sup>0</sup>*:*98*<sup>x</sup>*5,14 <sup>½</sup> ð Þ�þ 200 1*:*<sup>00</sup> � <sup>1</sup>*<sup>x</sup>*1,15 � <sup>0</sup>*:*87*<sup>x</sup>*2,15 � <sup>0</sup>*:*97*<sup>x</sup>*3,15 � <sup>0</sup>*:*98*<sup>x</sup>*4,15 � <sup>0</sup>*:*99*<sup>x</sup>*5,15 <sup>½</sup> ð Þ�þ 130 1*:*<sup>00</sup> � <sup>1</sup>*<sup>x</sup>*1,16 � <sup>0</sup>*:*88*<sup>x</sup>*2,16 � <sup>0</sup>*:*98*<sup>x</sup>*3,16 � <sup>0</sup>*:*93*<sup>x</sup>*4,16 � <sup>0</sup>*:*99*<sup>x</sup>*5,16 <sup>½</sup> ð Þ�þ

(7)

(8)

100 1*:*<sup>00</sup> � <sup>1</sup>*<sup>x</sup>*1,17 � <sup>0</sup>*:*85*<sup>x</sup>*2,18 � <sup>0</sup>*:*95*<sup>x</sup>*3,17 � <sup>1</sup>*<sup>x</sup>*4,17 � <sup>1</sup>*<sup>x</sup>*5,17 <sup>½</sup> ð Þ�þ 100 1*:*<sup>00</sup> � <sup>0</sup>*:*95*<sup>x</sup>*1,18 � <sup>0</sup>*:*84*<sup>x</sup>*2,18 � <sup>0</sup>*:*92*<sup>x</sup>*3,18 � <sup>1</sup>*<sup>x</sup>*4,18 � <sup>1</sup>*<sup>x</sup>*5,18 <sup>½</sup> ð Þ�þ 100 1*:*<sup>00</sup> � <sup>1</sup>*<sup>x</sup>*1,19 � <sup>0</sup>*:*85*<sup>x</sup>*2,19 � <sup>0</sup>*:*93*<sup>x</sup>*3,19 � <sup>1</sup>*<sup>x</sup>*4,19 � <sup>1</sup>*<sup>x</sup>*5,19 <sup>½</sup> ð Þ�þ 150 1*:*<sup>00</sup> � <sup>1</sup>*<sup>x</sup>*1,20 � <sup>0</sup>*:*85*<sup>x</sup>*2,20 � <sup>0</sup>*:*92*<sup>x</sup>*3,20 � <sup>1</sup>*<sup>x</sup>*4,20 � <sup>1</sup>*<sup>x</sup>*5,20 ½ � ð Þ

*x*<sup>11</sup> þ *x*<sup>12</sup> þ *x*<sup>13</sup> þ *x*<sup>14</sup> þ *x*<sup>15</sup> þ *x*<sup>16</sup> þ *x*<sup>17</sup> þ *x*<sup>18</sup> þ *x*<sup>19</sup> þ *x*1,10 þ *x*1,11 þ*x*1,12 þ *x*1,13 þ *x*1,14 þ *x*1,15 þ *x*1,16 þ *x*1,17 þ *x*1,18 þ *x*1,19 þ *x*1,20 ≤200 *x*<sup>21</sup> þ *x*<sup>22</sup> þ *x*<sup>23</sup> þ *x*<sup>24</sup> þ *x*<sup>25</sup> þ *x*<sup>26</sup> þ *x*<sup>27</sup> þ *x*<sup>28</sup> þ *x*<sup>29</sup> þ *x*2,10 þ *x*2,11 þ*x*2,12 þ *x*2,13 þ *x*2,14 þ *x*2,15 þ *x*2,16 þ *x*2,17 þ *x*2,18 þ *x*2,19 þ *x*2,20 ≤ 100 *x*<sup>31</sup> þ *x*<sup>32</sup> þ *x*<sup>33</sup> þ *x*<sup>34</sup> þ *x*<sup>35</sup> þ *x*<sup>36</sup> þ *x*<sup>37</sup> þ *x*<sup>38</sup> þ *x*<sup>39</sup> þ *x*3,10 þ *x*3,11 þ*x*3,12 þ *x*3,13 þ *x*3,14 þ *x*3,15 þ *x*3,16 þ *x*3,17 þ *x*3,18 þ *x*3,19 þ *x*3,20 ≤300 *x*<sup>41</sup> þ *x*<sup>42</sup> þ *x*<sup>43</sup> þ *x*<sup>44</sup> þ *x*<sup>45</sup> þ *x*<sup>46</sup> þ *x*<sup>47</sup> þ *x*<sup>48</sup> þ *x*<sup>49</sup> þ *x*4,10 þ *x*4,11 þ*x*4,12 þ *x*4,13 þ *x*4,14 þ *x*4,15 þ *x*4,16 þ *x*4,17 þ *x*4,18 þ *x*4,19 þ *x*4,20 ≤150

*x*<sup>51</sup> þ *x*<sup>52</sup> þ *x*<sup>53</sup> þ *x*<sup>54</sup> þ *x*<sup>55</sup> þ *x*<sup>56</sup> þ *x*<sup>57</sup> þ *x*<sup>58</sup> þ *x*<sup>59</sup> þ *x*5,10 þ *x*5,11 þ*x*5,12 þ *x*5,13 þ *x*5,14 þ *x*5,15 þ *x*5,16 þ *x*5,17 þ *x*5,18 þ *x*5,19 þ *x*5,20 ≤250

The linear constraints on the available number of weapons of the five types are,

*<sup>z</sup>* <sup>¼</sup> 60 1*:*<sup>00</sup> � <sup>1</sup>*x*<sup>11</sup> � <sup>0</sup>*:*84*x*<sup>21</sup> � <sup>0</sup>*:*96*x*<sup>31</sup> � <sup>1</sup>*x*<sup>41</sup> � <sup>0</sup>*:*92*x*<sup>51</sup> <sup>½</sup> ð Þ�þ 50 1*:*<sup>00</sup> � <sup>0</sup>*:*95*x*<sup>12</sup> � <sup>0</sup>*:*83*x*<sup>22</sup> � <sup>0</sup>*:*95*x*<sup>32</sup> � <sup>1</sup>*x*<sup>42</sup> � <sup>0</sup>*:*94*x*<sup>52</sup> <sup>½</sup> ð Þ�þ 50 1*:*<sup>00</sup> � <sup>1</sup>*<sup>x</sup>*<sup>13</sup> � <sup>0</sup>*:*85*<sup>x</sup>*<sup>23</sup> � <sup>0</sup>*:*96*<sup>x</sup>*<sup>33</sup> � <sup>1</sup>*<sup>x</sup>*<sup>43</sup> � <sup>0</sup>*:*92*<sup>x</sup>*<sup>53</sup> <sup>½</sup> ð Þ�þ 75 1*:*<sup>00</sup> � <sup>1</sup>*<sup>x</sup>*<sup>14</sup> � <sup>0</sup>*:*84*<sup>x</sup>*<sup>24</sup> � <sup>0</sup>*:*96*<sup>x</sup>*<sup>34</sup> � <sup>1</sup>*<sup>x</sup>*<sup>44</sup> � <sup>0</sup>*:*95*<sup>x</sup>*<sup>54</sup> <sup>½</sup> ð Þ�þ 40 1*:*<sup>00</sup> � <sup>1</sup>*<sup>x</sup>*<sup>15</sup> � <sup>0</sup>*:*85*<sup>x</sup>*<sup>25</sup> � <sup>0</sup>*:*96*<sup>x</sup>*<sup>35</sup> � <sup>1</sup>*<sup>x</sup>*<sup>45</sup> � <sup>0</sup>*:*95*<sup>x</sup>*<sup>55</sup> <sup>½</sup> ð Þ�þ 60 1*:*<sup>00</sup> � <sup>0</sup>*:*85*<sup>x</sup>*<sup>16</sup> � <sup>0</sup>*:*81*<sup>x</sup>*<sup>26</sup> � <sup>0</sup>*:*90*<sup>x</sup>*<sup>36</sup> � <sup>1</sup>*<sup>x</sup>*<sup>46</sup> � <sup>0</sup>*:*98*<sup>x</sup>*<sup>56</sup> <sup>½</sup> ð Þ�þ 35 1*:*<sup>00</sup> � <sup>0</sup>*:*90*<sup>x</sup>*<sup>17</sup> � <sup>0</sup>*:*81*<sup>x</sup>*<sup>27</sup> � <sup>0</sup>*:*92*<sup>x</sup>*<sup>37</sup> � <sup>1</sup>*<sup>x</sup>*<sup>47</sup> � <sup>0</sup>*:*98*<sup>x</sup>*<sup>57</sup> <sup>½</sup> ð Þ�þ 30 1*:*<sup>00</sup> � <sup>0</sup>*:*85*<sup>x</sup>*<sup>18</sup> � <sup>0</sup>*:*82*<sup>x</sup>*<sup>28</sup> � <sup>0</sup>*:*91*<sup>x</sup>*<sup>38</sup> � <sup>1</sup>*<sup>x</sup>*<sup>48</sup> � <sup>1</sup>*<sup>x</sup>*<sup>58</sup> <sup>½</sup> ð Þ�þ 25 1*:*<sup>00</sup> � <sup>0</sup>*:*80*<sup>x</sup>*<sup>19</sup> � <sup>0</sup>*:*80*<sup>x</sup>*<sup>29</sup> � <sup>0</sup>*:*92*<sup>x</sup>*<sup>39</sup> � <sup>1</sup>*<sup>x</sup>*<sup>49</sup> � <sup>1</sup>*<sup>x</sup>*<sup>59</sup> <sup>½</sup> ð Þ�þ

*Concepts, Applications and Emerging Opportunities in Industrial Engineering*

**Figure 3.** *Survival probabilities of targets by weapons.*

The number of available weapons and the military value of targets is shown in [19] **Figure 4**.

There are also some requirements for weapons to destroy particular targets. **Figure 5** shows the minimum number of weapons that must be assigned to some particular targets.

**Figure 4.** *Availability of weapons and target military values.*

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

#### **Figure 5.**

The number of available weapons and the military value of targets is shown in

There are also some requirements for weapons to destroy particular targets. **Figure 5** shows the minimum number of weapons that must be assigned to some

*Concepts, Applications and Emerging Opportunities in Industrial Engineering*

[19] **Figure 4**.

**Figure 3.**

**Figure 4.**

**192**

*Availability of weapons and target military values.*

particular targets.

*Survival probabilities of targets by weapons.*

*Minimum requirements of weapons assigned to targets.*

#### **3.7 Formulation of battle field example**

After having all the values of required parameters, we formulate the model corresponding to the given example for maximizing the total expected target damage value as follows:

*<sup>z</sup>* <sup>¼</sup> 60 1*:*<sup>00</sup> � <sup>1</sup>*x*<sup>11</sup> � <sup>0</sup>*:*84*x*<sup>21</sup> � <sup>0</sup>*:*96*x*<sup>31</sup> � <sup>1</sup>*x*<sup>41</sup> � <sup>0</sup>*:*92*x*<sup>51</sup> <sup>½</sup> ð Þ�þ 50 1*:*<sup>00</sup> � <sup>0</sup>*:*95*x*<sup>12</sup> � <sup>0</sup>*:*83*x*<sup>22</sup> � <sup>0</sup>*:*95*x*<sup>32</sup> � <sup>1</sup>*x*<sup>42</sup> � <sup>0</sup>*:*94*x*<sup>52</sup> <sup>½</sup> ð Þ�þ 50 1*:*<sup>00</sup> � <sup>1</sup>*<sup>x</sup>*<sup>13</sup> � <sup>0</sup>*:*85*<sup>x</sup>*<sup>23</sup> � <sup>0</sup>*:*96*<sup>x</sup>*<sup>33</sup> � <sup>1</sup>*<sup>x</sup>*<sup>43</sup> � <sup>0</sup>*:*92*<sup>x</sup>*<sup>53</sup> <sup>½</sup> ð Þ�þ 75 1*:*<sup>00</sup> � <sup>1</sup>*<sup>x</sup>*<sup>14</sup> � <sup>0</sup>*:*84*<sup>x</sup>*<sup>24</sup> � <sup>0</sup>*:*96*<sup>x</sup>*<sup>34</sup> � <sup>1</sup>*<sup>x</sup>*<sup>44</sup> � <sup>0</sup>*:*95*<sup>x</sup>*<sup>54</sup> <sup>½</sup> ð Þ�þ 40 1*:*<sup>00</sup> � <sup>1</sup>*<sup>x</sup>*<sup>15</sup> � <sup>0</sup>*:*85*<sup>x</sup>*<sup>25</sup> � <sup>0</sup>*:*96*<sup>x</sup>*<sup>35</sup> � <sup>1</sup>*<sup>x</sup>*<sup>45</sup> � <sup>0</sup>*:*95*<sup>x</sup>*<sup>55</sup> <sup>½</sup> ð Þ�þ 60 1*:*<sup>00</sup> � <sup>0</sup>*:*85*<sup>x</sup>*<sup>16</sup> � <sup>0</sup>*:*81*<sup>x</sup>*<sup>26</sup> � <sup>0</sup>*:*90*<sup>x</sup>*<sup>36</sup> � <sup>1</sup>*<sup>x</sup>*<sup>46</sup> � <sup>0</sup>*:*98*<sup>x</sup>*<sup>56</sup> <sup>½</sup> ð Þ�þ 35 1*:*<sup>00</sup> � <sup>0</sup>*:*90*<sup>x</sup>*<sup>17</sup> � <sup>0</sup>*:*81*<sup>x</sup>*<sup>27</sup> � <sup>0</sup>*:*92*<sup>x</sup>*<sup>37</sup> � <sup>1</sup>*<sup>x</sup>*<sup>47</sup> � <sup>0</sup>*:*98*<sup>x</sup>*<sup>57</sup> <sup>½</sup> ð Þ�þ 30 1*:*<sup>00</sup> � <sup>0</sup>*:*85*<sup>x</sup>*<sup>18</sup> � <sup>0</sup>*:*82*<sup>x</sup>*<sup>28</sup> � <sup>0</sup>*:*91*<sup>x</sup>*<sup>38</sup> � <sup>1</sup>*<sup>x</sup>*<sup>48</sup> � <sup>1</sup>*<sup>x</sup>*<sup>58</sup> <sup>½</sup> ð Þ�þ 25 1*:*<sup>00</sup> � <sup>0</sup>*:*80*<sup>x</sup>*<sup>19</sup> � <sup>0</sup>*:*80*<sup>x</sup>*<sup>29</sup> � <sup>0</sup>*:*92*<sup>x</sup>*<sup>39</sup> � <sup>1</sup>*<sup>x</sup>*<sup>49</sup> � <sup>1</sup>*<sup>x</sup>*<sup>59</sup> <sup>½</sup> ð Þ�þ 150 1*:*<sup>00</sup> � <sup>1</sup>*<sup>x</sup>*1,10 � <sup>0</sup>*:*86*<sup>x</sup>*2,10 � <sup>0</sup>*:*95*<sup>x</sup>*3,10 � <sup>0</sup>*:*96*<sup>x</sup>*4,10 � <sup>0</sup>*:*90*<sup>x</sup>*5,10 <sup>½</sup> ð Þ�þ 30 1*:*<sup>00</sup> � <sup>1</sup>*<sup>x</sup>*1,11 � <sup>1</sup>*<sup>x</sup>*2,11 � <sup>0</sup>*:*99*<sup>x</sup>*3,11 � <sup>0</sup>*:*91*<sup>x</sup>*4,11 � <sup>0</sup>*:*95*<sup>x</sup>*5,11 <sup>½</sup> ð Þ�þ 45 1*:*<sup>00</sup> � <sup>1</sup>*<sup>x</sup>*1,12 � <sup>0</sup>*:*98*<sup>x</sup>*2,12 � <sup>0</sup>*:*98*<sup>x</sup>*3,12 � <sup>0</sup>*:*92*<sup>x</sup>*4,12 � <sup>0</sup>*:*96*<sup>x</sup>*5,12 <sup>½</sup> ð Þ�þ 125 1*:*<sup>00</sup> � <sup>1</sup>*<sup>x</sup>*1,13 � <sup>1</sup>*<sup>x</sup>*2,13 � <sup>0</sup>*:*99*<sup>x</sup>*3,13 � <sup>0</sup>*:*91*<sup>x</sup>*4,13 � <sup>0</sup>*:*91*<sup>x</sup>*5,13 <sup>½</sup> ð Þ�þ 200 1*:*<sup>00</sup> � <sup>1</sup>*<sup>x</sup>*1,14 � <sup>0</sup>*:*88*<sup>x</sup>*2,14 � <sup>0</sup>*:*98*<sup>x</sup>*3,14 � <sup>0</sup>*:*92*<sup>x</sup>*4,14 � <sup>0</sup>*:*98*<sup>x</sup>*5,14 <sup>½</sup> ð Þ�þ 200 1*:*<sup>00</sup> � <sup>1</sup>*<sup>x</sup>*1,15 � <sup>0</sup>*:*87*<sup>x</sup>*2,15 � <sup>0</sup>*:*97*<sup>x</sup>*3,15 � <sup>0</sup>*:*98*<sup>x</sup>*4,15 � <sup>0</sup>*:*99*<sup>x</sup>*5,15 <sup>½</sup> ð Þ�þ 130 1*:*<sup>00</sup> � <sup>1</sup>*<sup>x</sup>*1,16 � <sup>0</sup>*:*88*<sup>x</sup>*2,16 � <sup>0</sup>*:*98*<sup>x</sup>*3,16 � <sup>0</sup>*:*93*<sup>x</sup>*4,16 � <sup>0</sup>*:*99*<sup>x</sup>*5,16 <sup>½</sup> ð Þ�þ 100 1*:*<sup>00</sup> � <sup>1</sup>*<sup>x</sup>*1,17 � <sup>0</sup>*:*85*<sup>x</sup>*2,18 � <sup>0</sup>*:*95*<sup>x</sup>*3,17 � <sup>1</sup>*<sup>x</sup>*4,17 � <sup>1</sup>*<sup>x</sup>*5,17 <sup>½</sup> ð Þ�þ 100 1*:*<sup>00</sup> � <sup>0</sup>*:*95*<sup>x</sup>*1,18 � <sup>0</sup>*:*84*<sup>x</sup>*2,18 � <sup>0</sup>*:*92*<sup>x</sup>*3,18 � <sup>1</sup>*<sup>x</sup>*4,18 � <sup>1</sup>*<sup>x</sup>*5,18 <sup>½</sup> ð Þ�þ 100 1*:*<sup>00</sup> � <sup>1</sup>*<sup>x</sup>*1,19 � <sup>0</sup>*:*85*<sup>x</sup>*2,19 � <sup>0</sup>*:*93*<sup>x</sup>*3,19 � <sup>1</sup>*<sup>x</sup>*4,19 � <sup>1</sup>*<sup>x</sup>*5,19 <sup>½</sup> ð Þ�þ 150 1*:*<sup>00</sup> � <sup>1</sup>*<sup>x</sup>*1,20 � <sup>0</sup>*:*85*<sup>x</sup>*2,20 � <sup>0</sup>*:*92*<sup>x</sup>*3,20 � <sup>1</sup>*<sup>x</sup>*4,20 � <sup>1</sup>*<sup>x</sup>*5,20 ½ � ð Þ (7)

#### subject to,

The linear constraints on the available number of weapons of the five types are,

*x*<sup>11</sup> þ *x*<sup>12</sup> þ *x*<sup>13</sup> þ *x*<sup>14</sup> þ *x*<sup>15</sup> þ *x*<sup>16</sup> þ *x*<sup>17</sup> þ *x*<sup>18</sup> þ *x*<sup>19</sup> þ *x*1,10 þ *x*1,11 þ*x*1,12 þ *x*1,13 þ *x*1,14 þ *x*1,15 þ *x*1,16 þ *x*1,17 þ *x*1,18 þ *x*1,19 þ *x*1,20 ≤200 *x*<sup>21</sup> þ *x*<sup>22</sup> þ *x*<sup>23</sup> þ *x*<sup>24</sup> þ *x*<sup>25</sup> þ *x*<sup>26</sup> þ *x*<sup>27</sup> þ *x*<sup>28</sup> þ *x*<sup>29</sup> þ *x*2,10 þ *x*2,11 þ*x*2,12 þ *x*2,13 þ *x*2,14 þ *x*2,15 þ *x*2,16 þ *x*2,17 þ *x*2,18 þ *x*2,19 þ *x*2,20 ≤ 100 *x*<sup>31</sup> þ *x*<sup>32</sup> þ *x*<sup>33</sup> þ *x*<sup>34</sup> þ *x*<sup>35</sup> þ *x*<sup>36</sup> þ *x*<sup>37</sup> þ *x*<sup>38</sup> þ *x*<sup>39</sup> þ *x*3,10 þ *x*3,11 þ*x*3,12 þ *x*3,13 þ *x*3,14 þ *x*3,15 þ *x*3,16 þ *x*3,17 þ *x*3,18 þ *x*3,19 þ *x*3,20 ≤300 *x*<sup>41</sup> þ *x*<sup>42</sup> þ *x*<sup>43</sup> þ *x*<sup>44</sup> þ *x*<sup>45</sup> þ *x*<sup>46</sup> þ *x*<sup>47</sup> þ *x*<sup>48</sup> þ *x*<sup>49</sup> þ *x*4,10 þ *x*4,11 þ*x*4,12 þ *x*4,13 þ *x*4,14 þ *x*4,15 þ *x*4,16 þ *x*4,17 þ *x*4,18 þ *x*4,19 þ *x*4,20 ≤150 *x*<sup>51</sup> þ *x*<sup>52</sup> þ *x*<sup>53</sup> þ *x*<sup>54</sup> þ *x*<sup>55</sup> þ *x*<sup>56</sup> þ *x*<sup>57</sup> þ *x*<sup>58</sup> þ *x*<sup>59</sup> þ *x*5,10 þ *x*5,11 þ*x*5,12 þ *x*5,13 þ *x*5,14 þ *x*5,15 þ *x*5,16 þ *x*5,17 þ *x*5,18 þ *x*5,19 þ *x*5,20 ≤250 (8)

And the linear constraints on the minimum required assignment of weapons to the seven specified targets that must be engaged are:

$$\begin{aligned} & \mathbf{x}\_{11} + \mathbf{x}\_{21} + \mathbf{x}\_{31} + \mathbf{x}\_{41} + \mathbf{x}\_{51} \ge 30 \\ & \mathbf{x}\_{16} + \mathbf{x}\_{26} + \mathbf{x}\_{36} + \mathbf{x}\_{46} + \mathbf{x}\_{56} \ge 100 \\ & \mathbf{x}\_{1,10} + \mathbf{x}\_{2,10} + \mathbf{x}\_{3,10} + \mathbf{x}\_{4,10} + \mathbf{x}\_{5,10} \ge 40 \\ & \mathbf{x}\_{1,14} + \mathbf{x}\_{2,14} + \mathbf{x}\_{3,14} + \mathbf{x}\_{4,14} + \mathbf{x}\_{5,14} \ge 50 \\ & \mathbf{x}\_{1,15} + \mathbf{x}\_{2,15} + \mathbf{x}\_{3,15} + \mathbf{x}\_{4,15} + \mathbf{x}\_{5,15} \ge 70 \\ & \mathbf{x}\_{1,16} + \mathbf{x}\_{2,16} + \mathbf{x}\_{3,16} + \mathbf{x}\_{4,16} + \mathbf{x}\_{5,16} \ge 35 \\ & \mathbf{x}\_{1,20} + \mathbf{x}\_{2,20} + \mathbf{x}\_{3,20} + \mathbf{x}\_{4,20} + \mathbf{x}\_{5,20} \ge 30 \end{aligned} \tag{9}$$

X *J*

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

*Weapon Target Assignment*

*u j* ½ � <sup>∗</sup> <sup>1</sup> � <sup>Y</sup>

*I*

<sup>1</sup> � *<sup>p</sup>ij*

*x i*½ � , *j* ≤ *w i*½ �; *i* ¼ 1, … , *M*;

*x i*½ � , *j* ≤ � *t j* ½ �; *i* ¼ 1, … , *N*;

*Step VII: Then run this code in the 'run' file to calculate the objective function value that is to be maximized by using the solver option such as MINOS, BARON, BONMIN, MINLP, and CONOPT. By using the command "EXPAND" we can show the expan-*

Using the new developed algorithm by AMPL, we can solve the WTAP for the large numbers of weapons and targets using the single model file with different data

As our developed method is based on computerized tools, so we first develop the general code in AMPL. Then update the data file for the Eqs. (7)–(9). And finally run the AMPL code, then we get our desired result as an output file (Appendix-A) in AMPL. Subsequent to adjusting the quantity of weapons to the closest whole

For several years this type of weapon-target assignment has been performed at the Research analysis. Here we have taken the numerical problem presented in [18]. We have presented the result of the WTAP by using our developed method in **Table 2**. Bracken et al. [18] solved this problem and got a set of solution of the

*kill*½ � *i*, *j* � � <sup>∧</sup> *x i*½ � , *<sup>j</sup>* !

*i*¼1

*j*¼1

*Step V: Define a set of equations of constraints,*

X *J*

*j*¼1

� X *I*

*sion of objective function and constraints in the console. Step VIII: Display the solution value in the console.*

values according to the different scale problems.

numbers, the outcomes have appeared in.

**Figure 6.**

**195**

*Number of weapons assigned to targets.*

**3.11 Results of the WTAP using our developed method**

**3.12 Comparison between two results of the WTAP**

number of weapons assigned to targets shown in **Figure 6**.

*i*¼1

*Step VI: Input data of the defined parameters in the 'dat' file.*

## **3.8 Computational complexity of WTAP**

The general WTA problem is the situation where a number of *W*weapon systems have to engage a number of *T*targets. All weapon systems and all targets may have different characteristics. Also, different weapon systems may require a different amount of time to engage a target. When*T* > >*W* an additional problem occurs. So as the scale of WTA problem grows, its computational requirement grows exponentially. So it is quite impossible to solve this type of large scale WTA problem directly. So, computational algorithms are the best approach to solve the large scale dynamic WTA problem [8].

#### **3.9 Our solution approach for the WTA model**

After formulating the problem, we have the Eqs. (7)–(9). We observe that we have total 100 variables with a nonlinear exponential objective function and 12 linear constraints, which is quite large. There does not exist any exact methods for the WTA problem even relatively small size problems. As there are so many computer based software tools to solve different types of mathematical problems, we propose a computer oriented algorithm to solve such large scale problems in a short time. Our proposed algorithm not only solve large scale WTA problems but also small problems in a single framework. We develop a computerized algorithm in which all types of target-based WTA problem can be solved in a reasonably fast time to help decision makers to make proper assignment on the battlefield.

#### **3.10 The structure of our proposed algorithm for solving the WTAP**

Since no real time exact solutions to WTAs are available, either for static or dynamic versions, alternative approximation methodologies must be considered, including heuristic techniques. We develop our computerized algorithm by using the Mathematical Programming Language AMPL.

**Algorithm**: Our developed Algorithm in AMPL.

*Step I: Initialize parameters N, M > 0 and set integers I, J, K. Step II: Input number of weapons (M) and number of targets (N) and introduce non-negative integer variablesx i* f g , *j .*

*Step III: Introduce the parameters wi* ≥0, *tj* ≥0, *u <sup>j</sup>* ≥0, 0 ≤*pij kill* ≤ 1*:. Step IV: Define the non-convex objective function to maximize*

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

And the linear constraints on the minimum required assignment of weapons to

*x*<sup>11</sup> þ *x*<sup>21</sup> þ *x*<sup>31</sup> þ *x*<sup>41</sup> þ *x*<sup>51</sup> ≥30

*Concepts, Applications and Emerging Opportunities in Industrial Engineering*

*x*<sup>16</sup> þ *x*<sup>26</sup> þ *x*<sup>36</sup> þ *x*<sup>46</sup> þ *x*<sup>56</sup> ≥100

*x*1,10 þ *x*2,10 þ *x*3,10 þ *x*4,10 þ *x*5,10 ≥ 40

*x*1,14 þ *x*2,14 þ *x*3,14 þ *x*4,14 þ *x*5,14 ≥50

(9)

*x*1,15 þ *x*2,15 þ *x*3,15 þ *x*4,15 þ *x*5,15 ≥70

*x*1,16 þ *x*2,16 þ *x*3,16 þ *x*4,16 þ *x*5,16 ≥35

*x*1,20 þ *x*2,20 þ *x*3,20 þ *x*4,20 þ *x*5,20 ≥30

The general WTA problem is the situation where a number of *W*weapon systems have to engage a number of *T*targets. All weapon systems and all targets may have different characteristics. Also, different weapon systems may require a different amount of time to engage a target. When*T* > >*W* an additional problem occurs. So as the scale of WTA problem grows, its computational requirement grows exponentially. So it is quite impossible to solve this type of large scale WTA problem directly. So, computational algorithms are the best approach to solve the large scale

After formulating the problem, we have the Eqs. (7)–(9). We observe that we have total 100 variables with a nonlinear exponential objective function and 12 linear constraints, which is quite large. There does not exist any exact methods for the WTA problem even relatively small size problems. As there are so many computer based software tools to solve different types of mathematical problems, we propose a computer oriented algorithm to solve such large scale problems in a short time. Our proposed algorithm not only solve large scale WTA problems but also small problems in a single framework. We develop a computerized algorithm in which all types of target-based WTA problem can be solved in a reasonably fast time to help decision makers to make proper assignment on the battlefield.

**3.10 The structure of our proposed algorithm for solving the WTAP**

Since no real time exact solutions to WTAs are available, either for static or dynamic versions, alternative approximation methodologies must be considered, including heuristic techniques. We develop our computerized algorithm by using

*Step II: Input number of weapons (M) and number of targets (N) and introduce*

*kill* ≤ 1*:.*

the seven specified targets that must be engaged are:

**3.8 Computational complexity of WTAP**

**3.9 Our solution approach for the WTA model**

the Mathematical Programming Language AMPL.

*non-negative integer variablesx i* f g , *j .*

**194**

**Algorithm**: Our developed Algorithm in AMPL.

*Step I: Initialize parameters N, M > 0 and set integers I, J, K.*

*Step III: Introduce the parameters wi* ≥0, *tj* ≥0, *u <sup>j</sup>* ≥0, 0 ≤*pij*

*Step IV: Define the non-convex objective function to maximize*

dynamic WTA problem [8].

$$\sum\_{j=1}^{J} \mu[j] \ast \left( \mathbf{1} - \prod\_{i=1}^{I} \left( \mathbf{1} - p\_{kill}^{ij}[i, j] \right) \wedge \mathbf{x}[i, j] \right)$$

*Step V: Define a set of equations of constraints,*

$$\sum\_{j=1}^{J} \mathbf{x}[i, j] \le w[i]; \ i = \mathbf{1}, \ \dots, M;$$

$$-\sum\_{i=1}^{I} \mathbf{x}[i, j] \le -t[j]; \ i = \mathbf{1}, \ \dots, N;$$

*Step VI: Input data of the defined parameters in the 'dat' file.*

*Step VII: Then run this code in the 'run' file to calculate the objective function value that is to be maximized by using the solver option such as MINOS, BARON, BONMIN, MINLP, and CONOPT. By using the command "EXPAND" we can show the expansion of objective function and constraints in the console. Step VIII: Display the solution value in the console.*

Using the new developed algorithm by AMPL, we can solve the WTAP for the large numbers of weapons and targets using the single model file with different data values according to the different scale problems.

#### **3.11 Results of the WTAP using our developed method**

As our developed method is based on computerized tools, so we first develop the general code in AMPL. Then update the data file for the Eqs. (7)–(9). And finally run the AMPL code, then we get our desired result as an output file (Appendix-A) in AMPL. Subsequent to adjusting the quantity of weapons to the closest whole numbers, the outcomes have appeared in.

#### **3.12 Comparison between two results of the WTAP**

For several years this type of weapon-target assignment has been performed at the Research analysis. Here we have taken the numerical problem presented in [18]. We have presented the result of the WTAP by using our developed method in **Table 2**. Bracken et al. [18] solved this problem and got a set of solution of the number of weapons assigned to targets shown in **Figure 6**.

**Figure 6.** *Number of weapons assigned to targets.*

night time of the day. Also, they take 15 vehicles such as somoy news, BTV, Channel I, NTV, ETV, ATN News, GTV, Radio Today, Radio Foorti, Facebook, Prothom Alo, Ittefaq, Billboard, Printings, and E-mail. That company knows the percentages of reaching the target audiences in different time partitions according to the mentioned media vehicles. The probabilities of reaching target audiences are shown in the following table. In **Table 2**, we can see that some vehicles have 0 probability to reach some targets. Prime time is the most important segment, as night time is the least important segment for the product. Moreover, the segment weights facilitate marketers to give relative importance with respect to product or service characteristics.

Our objective is to make a proper assignment of ads to targets for maximizing the effectiveness of advertising. The objective function along with total 19 constraints (15 supply constraints for media vehicles and 4 demand constraints for

(10)

(11)

The weights can be changed with respect to the features of the product.

<sup>2</sup>½1*:*<sup>00</sup> � ð0*:*79*<sup>x</sup>*<sup>11</sup> � <sup>0</sup>*:*65*<sup>x</sup>*<sup>21</sup> � <sup>0</sup>*:*81*<sup>x</sup>*<sup>31</sup> � <sup>1</sup>*<sup>x</sup>*<sup>41</sup> � <sup>0</sup>*:*87*<sup>x</sup>*<sup>51</sup> � <sup>0</sup>*:*76*<sup>x</sup>*<sup>61</sup> � <sup>0</sup>*:*91*<sup>x</sup>*<sup>71</sup> �

<sup>þ</sup>4½1*:*<sup>00</sup> � ð0*:*88*<sup>x</sup>*<sup>13</sup> � <sup>0</sup>*:*88*<sup>x</sup>*<sup>23</sup> � <sup>1</sup>*<sup>x</sup>*<sup>33</sup> � <sup>0</sup>*:*81*<sup>x</sup>*<sup>43</sup> � <sup>0</sup>*:*75*<sup>x</sup>*<sup>53</sup> � <sup>0</sup>*:*78*<sup>x</sup>*<sup>63</sup> � <sup>0</sup>*:*82*<sup>x</sup>*<sup>73</sup> � <sup>0</sup>*:*53*<sup>x</sup>*<sup>83</sup> � <sup>0</sup>*:*86*<sup>x</sup>*<sup>93</sup> � <sup>0</sup>*:*97*<sup>x</sup>*10,3 � <sup>0</sup>*:*97*<sup>x</sup>*11,3 � <sup>0</sup>*:*91*<sup>x</sup>*12,3 � <sup>0</sup>*:*72*<sup>x</sup>*13,3 � <sup>0</sup>*:*92*<sup>x</sup>*14,3 � <sup>0</sup>*:*96*<sup>x</sup>*15,3 Þ� <sup>þ</sup>1½1*:*<sup>00</sup> � ð0*:*77*<sup>x</sup>*<sup>14</sup> � <sup>0</sup>*:*93*<sup>x</sup>*<sup>24</sup> � <sup>0</sup>*:*81*<sup>x</sup>*<sup>34</sup> � <sup>0</sup>*:*87*<sup>x</sup>*<sup>44</sup> � <sup>1</sup>*<sup>x</sup>*<sup>54</sup> � <sup>0</sup>*:*81*<sup>x</sup>*<sup>64</sup> � <sup>0</sup>*:*72*<sup>x</sup>*<sup>74</sup> � <sup>1</sup>*<sup>x</sup>*<sup>84</sup> � <sup>0</sup>*:*57*<sup>x</sup>*<sup>94</sup> � <sup>0</sup>*:*65*<sup>x</sup>*10,4 � <sup>0</sup>*:*91*<sup>x</sup>*11,4 � <sup>0</sup>*:*79*<sup>x</sup>*12,4 � <sup>0</sup>*:*98*<sup>x</sup>*13,4 � <sup>0</sup>*:*97*<sup>x</sup>*14,4 � <sup>0</sup>*:*68*<sup>x</sup>*15,4 Þ�

<sup>0</sup>*:*61*<sup>x</sup>*<sup>81</sup> � <sup>0</sup>*:*76*<sup>x</sup>*<sup>91</sup> � <sup>0</sup>*:*9*<sup>x</sup>*10,1 � <sup>0</sup>*:*88*<sup>x</sup>*11,1 � <sup>0</sup>*:*68*<sup>x</sup>*12,1 � <sup>0</sup>*:*68*<sup>x</sup>*13,1 � <sup>0</sup>*:*77*<sup>x</sup>*14,1 � <sup>0</sup>*:*71*<sup>x</sup>*15,1 Þ� <sup>þ</sup>3½1*:*<sup>00</sup> � ð0*:*88*<sup>x</sup>*<sup>12</sup> � <sup>0</sup>*:*76*<sup>x</sup>*<sup>22</sup> � <sup>0</sup>*:*96*<sup>x</sup>*<sup>32</sup> � <sup>0</sup>*:*74*<sup>x</sup>*<sup>42</sup> � <sup>0</sup>*:*81*<sup>x</sup>*<sup>52</sup> � <sup>0</sup>*:*86*<sup>x</sup>*<sup>62</sup> � <sup>1</sup>*<sup>x</sup>*<sup>72</sup> � <sup>0</sup>*:*83*<sup>x</sup>*<sup>82</sup> � <sup>0</sup>*:*69*<sup>x</sup>*<sup>92</sup> � <sup>0</sup>*:*77*<sup>x</sup>*10,2 � <sup>0</sup>*:*89*<sup>x</sup>*11,2 � <sup>0</sup>*:*77*<sup>x</sup>*12,2 � <sup>0</sup>*:*90*<sup>x</sup>*13,2 � <sup>0</sup>*:*88*<sup>x</sup>*14,2 � <sup>0</sup>*:*93*<sup>x</sup>*15,2 Þ�

Subject to, the linear constraints on the available number ads of 15 media

*x*<sup>11</sup> þ *x*<sup>12</sup> þ *x*<sup>13</sup> þ *x*<sup>14</sup> ≤8 *x*<sup>21</sup> þ *x*<sup>22</sup> þ *x*<sup>23</sup> þ *x*<sup>24</sup> ≤ 7 *x*<sup>31</sup> þ *x*<sup>32</sup> þ *x*<sup>33</sup> þ *x*<sup>34</sup> ≤9 *x*<sup>41</sup> þ *x*<sup>42</sup> þ *x*<sup>43</sup> þ *x*<sup>44</sup> ≤ 5 *x*<sup>51</sup> þ *x*<sup>52</sup> þ *x*<sup>53</sup> þ *x*<sup>54</sup> ≤6 *x*<sup>61</sup> þ *x*<sup>62</sup> þ *x*<sup>63</sup> þ *x*<sup>64</sup> ≤8 *x*<sup>71</sup> þ *x*<sup>72</sup> þ *x*<sup>73</sup> þ *x*<sup>74</sup> ≤3 *x*<sup>81</sup> þ *x*<sup>82</sup> þ *x*<sup>83</sup> þ *x*<sup>84</sup> ≤10 *x*<sup>91</sup> þ *x*<sup>92</sup> þ *x*<sup>93</sup> þ *x*<sup>94</sup> ≤15 *x*10,1 þ *x*10,2 þ *x*10,3 þ *x*10,4 ≤12 *x*11,1 þ *x*11,2 þ *x*11,3 þ *x*11,4 ≤8 *x*12,1 þ *x*12,2 þ *x*12,3 þ *x*12,4 ≤ 4 *x*13,1 þ *x*13,2 þ *x*13,3 þ *x*13,4 ≤4 *x*14,1 þ *x*14,2 þ *x*14,3 þ *x*14,4 ≤ 4 *x*15,1 þ *x*15,2 þ *x*15,3 þ *x*15,4 ≤4

*kill*):

**3.14 Formulation of media allocation problem**

Probability Matrix (*pij*

*Weapon Target Assignment*

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

target audiences) are given below:

Maximize, z =

types are,

**197**

#### **Figure 7.**

*Comparison of the results between the two methods.*

Now to check the efficiency of our model we compare the two results of the problem graphically. Then for the both results we calculate the objective function that is to be maximized. Here the graphical representation of total number of weapons assigned to targets of the results are shown in **Figure 7**.

**Objective Function (Existing Solution):** Max *z* ¼ 1733*:*81

**Objective Function (Our Result):** Max *z* ¼ 1735*:*57

Comparing the above two results, we have the better result than the existing result. That is, we have the maximum objective function. This concludes that our proposed method gives the effective result. Our developed AMPL code studied in this Chapter improved the existing solution by 0.1%.

## **3.13 Numerical example of media allocation**

Comparing media allocation with the WTA problem, we can consider the weapons as media vehicles to be advertised when the military targets as target audiences to be intended to reach. People exposed by media vehicles at different times of the day are given as target audiences. The weapon numbers *xij* are determined as the number of ads.

The number of ads refers to the number of times within a given period time an audience is exposed to a media schedule. The mathematical programming model is as follows under the assumption that the target audience is constant to be exposed by such media vehicles in given period time.

We formulate the media allocation as the weapon-target assignment model which satisfies the Eqs. (3)–(6).

Here *i* ¼ 1, 2, … ,*W* be the number of kinds of advertisements,

*j* ¼ 1, 2, … , *T* be the number of segments,

*wi* be the available number of advertisements of type *i*,

*tj* be the minimum required number of ads for the target audience *j* s,

*u <sup>j</sup>* be the relative segment weights.

*xij* be the number of advertisements of type *i* assigned to target *j*,

*pij kill* be the probability of reaching the target audience *j* by a single ad type *i*.

So here the objective is to maximize the total probability of reaching the target audiences.

Suppose a company is planning to start an advertising campaign for a particular product. That company takes four target audiences as morning, afternoon, prime and *Weapon Target Assignment DOI: http://dx.doi.org/10.5772/intechopen.93665*

night time of the day. Also, they take 15 vehicles such as somoy news, BTV, Channel I, NTV, ETV, ATN News, GTV, Radio Today, Radio Foorti, Facebook, Prothom Alo, Ittefaq, Billboard, Printings, and E-mail. That company knows the percentages of reaching the target audiences in different time partitions according to the mentioned media vehicles. The probabilities of reaching target audiences are shown in the following table. In **Table 2**, we can see that some vehicles have 0 probability to reach some targets. Prime time is the most important segment, as night time is the least important segment for the product. Moreover, the segment weights facilitate marketers to give relative importance with respect to product or service characteristics. The weights can be changed with respect to the features of the product.

Probability Matrix (*pij kill*):
