**3.14 Formulation of media allocation problem**

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 target audiences) are given below:

Maximize, z =

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

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

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

We formulate the media allocation as the weapon-target assignment model

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

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

*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

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

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

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

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

*Concepts, Applications and Emerging Opportunities in Industrial Engineering*

this Chapter improved the existing solution by 0.1%.

**3.13 Numerical example of media allocation**

*Comparison of the results between the two methods.*

by such media vehicles in given period time.

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

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

which satisfies the Eqs. (3)–(6).

*pij*

**Figure 7.**

audiences.

**196**

<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>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� <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� (10)

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

$$\begin{aligned} &x\_{11} + x\_{12} + x\_{13} + x\_{14} \le 8 \\ &x\_{21} + x\_{22} + x\_{23} + x\_{24} \le 7 \\ &x\_{31} + x\_{32} + x\_{33} + x\_{34} \le 9 \\ &x\_{41} + x\_{42} + x\_{43} + x\_{44} \le 5 \\ &x\_{51} + x\_{52} + x\_{53} + x\_{54} \le 6 \\ &x\_{61} + x\_{62} + x\_{63} + x\_{64} \le 8 \\ &x\_{71} + x\_{72} + x\_{73} + x\_{74} \le 3 \\ &x\_{81} + x\_{82} + x\_{83} + x\_{84} \le 10 \\ &x\_{91} + x\_{92} + x\_{93} + x\_{94} \le 15 \\ &x\_{10,1} + x\_{10,2} + x\_{10,3} + x\_{10,4} \le 12 \\ &x\_{11,1} + x\_{11,2} + x\_{11,3} + x\_{11,4} \le 8 \\ &x\_{12,1} + x\_{12,2} + x\_{12,3} + x\_{13,4} \le 4 \\ &x\_{13,1} + x\_{13,2} + x\_{13,3} + x\_{14,4} \le 4 \\ &x\_{14,1} + x\_{14,2} + x\_{14,3} + x\_{14,4} \le 4 \\ &x\_{15,1} + x\_{15,2} + x\_{15,3} + x\_{15,4} \le 4 \end{aligned}$$


#### *Concepts, Applications and Emerging Opportunities in Industrial Engineering*

#### **Table 2.**

*The probability of reaching target audiences.*

And, the linear constraints on the minimum required ads of media vehicles to the four specified target audiences that must be engaged are:

*x*<sup>11</sup> þ *x*<sup>21</sup> þ *x*<sup>31</sup> þ *x*<sup>41</sup> þ *x*<sup>51</sup> þ *x*<sup>61</sup> þ *x*<sup>71</sup> þ *x*81þ *x*<sup>91</sup> þ *x*10,1 þ *x*11,1 þ *x*12,1 þ *x*13,1 þ *x*14,1 þ *x*15,1 ≥16 *x*<sup>12</sup> þ *x*<sup>22</sup> þ *x*<sup>32</sup> þ *x*<sup>42</sup> þ *x*<sup>52</sup> þ *x*<sup>62</sup> þ *x*<sup>72</sup> þ *x*82þ *x*<sup>92</sup> þ *x*10,2 þ *x*11,2 þ *x*12,2 þ *x*13,2 þ *x*14,2 þ *x*15,2 ≥18 *x*<sup>13</sup> þ *x*<sup>23</sup> þ *x*<sup>33</sup> þ *x*<sup>43</sup> þ *x*<sup>53</sup> þ *x*<sup>63</sup> þ *x*<sup>73</sup> þ *x*83þ *x*<sup>93</sup> þ *x*10,3 þ *x*11,3 þ *x*12,3 þ *x*13,3 þ *x*14,3 þ *x*15,3 ≥25 *x*<sup>14</sup> þ *x*<sup>24</sup> þ *x*<sup>34</sup> þ *x*<sup>44</sup> þ *x*<sup>54</sup> þ *x*<sup>64</sup> þ *x*<sup>74</sup> þ *x*84þ *x*<sup>94</sup> þ *x*10,4 þ *x*11,4 þ *x*12,4 þ *x*13,4 þ *x*14,4 þ *x*15,4 ≥ 10 (12)

**3.16 Comparison of media allocation result with other existing solution**

**Figure 8.**

**Table 3.**

**199**

*Media allocation solution by genetic algorithm.*

*Number of ads reaching to target audiences.*

*Weapon Target Assignment*

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

This hypothetical example was given and solved by using MS Excel [20] and meta-heuristic genetic algorithm [21] previously. We have used our proposed algorithm to solve the media allocation problem. The solutions obtained by using genetic algorithm [21] and MS Excel [20] are shown in **Tables 3** and **4**, respectively [22]. To check the efficiency of our model, we need to calculate the objective function for all the existing solution that is to be maximized. So the graphical representation

**Media vehicles Morning time (1) Afternoon time (2) Prime time (3) Night time (4)** Somoy News (1) 0 0 0 0 BTV (2) 7 0 0 0 Channel I (3) 6 1 0 2 NTV (4) 0 5 0 0 ETV (5) 0 0 6 0 ATN News (6) 2 0 5 0 GTV (7) 0 0 0 0 Radio Today (8) 2 0 8 0 Radio Foorti (9) 0 14 0 1 Facebook (10) 0 0 0 12 Prothom Alo (11) 2 2 2 2 Ittefaq (12) 1 1 1 1 Billboard (13) 1 1 1 1 Printings (14) 1 1 1 1 E-mail (15) 1 1 1 1 Total no of ads 23 26 25 21

#### **3.15 Solution of the media allocation problem**

We develop a near optimization model which allocate media vehicles to predetermined target segments. As this media allocation problem is formulated by using weapon-target assignment problem with 60 decision variables. By using our algorithm, we have solved the Media Allocation problem in a short time. In this case, we only change the data values in the 'dat' file, use the same mod.file and run. file. The result is given in **Figure 8**.
