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Question

An advertising company wishes to plan its advertising strategy in three different media-television,radio and magazines.The purpose of advertising is to reach as large a number of potential customers as possible.Following data have been obtained from market survey:
StrategiesTelevisionRadioMagazine-IMagazine-II
Cost of an advertising companyRs.$$30,000$$Rs.$$20,000$$Rs.$$15,000$$Rs.$$10,000$$
No.of potential customers reached per unit$$2,00,000$$$$6,00,000$$$$1,50,000$$$$1,00,000$$
No.of female customers reached per unit$$1,50,000$$$$4,00,000$$$$70,000$$$$50,000$$
The company wants to spend not more than Rs.$$4,50,000$$ on advertising.Following are the further requirements that must be met:
$$\left(i\right)$$at least $$1$$ million exposures take place among female customers.
$$\left(ii\right)$$advertising on magazines be limited to Rs.$$1,50,000$$
$$\left(iii\right)$$at least $$3$$ advertising units be bought on magazine $$I$$ and $$2$$ units on magazine $$II$$
$$\left(iv\right)$$the number of advertising units on television and radio should each be between $$5$$ and $$10.$$
Formulate an L.P model for the problem.


Solution

Formulation of Linear programming problem:
Let $${x}_{1},{x}_{2},{x}_{3}$$ and $${x}_{4}$$ denote the number of advertising units to be bought on television,radio,magazine$$I$$ and magazine$$II$$ respectively.
The objective is to maximize the total number of potential customers reached.
i.e., Maximize $$Z={10}^{5}\left(2{x}_{1}+6{x}_{2}+1.5{x}_{3}+{x}_{4}\right)$$
Constraints are 
on the advertising budget: $$30,000{x}_{1}+20,000{x}_{2}+15,000{x}_{3}+10,000{x}_{4}\le 4,50,000$$ 
or $$30{x}_{1}+20{x}_{2}+15{x}_{3}+10{x}_{4}\le 450$$
on the number of female customers reached by the advertising campaign:
$$1,50,000{x}_{1}+4,00,000{x}_{2}+70,000{x}_{3}+50,000{x}_{4}\ge 10,00,000$$
or $$15{x}_{1}+40{x}_{2}+7{x}_{3}+5{x}_{4}\ge100.$$
on expenses on magazine advertising:$$15,000{x}_{3}+10,000{x}_{4}\le1,50,000$$
or $$15{x}_{3}+10{x}_{4}\le 150$$
on no. of units on magazines:$${x}_{3}\ge3,{x}_{4}\ge2$$
on no. of units on television:$$5\le{x}_{1}\le10$$ or $${x}_{1}\ge5,{x}_{1}\le10,$$
       on no. of units on radio:$$5\le{x}_{2}\le10$$ or $${x}_{2}\ge5,{x}_{2}\le10,$$
where $${x}_{1},{x}_{2},{x}_{3},{x}_{4},$$each $$\ge0.$$

Mathematics

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