document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1166337>Question 1166337</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A company has $14,830 available per month for advertising. Newspaper ads cost $190 each and can't run more than 24 times per month. Radio ads cost $590 each and can't run more than 32 times per month at this price.\r<BR>\n" );
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document.write( "Each newspaper ad reaches 5700 potential customers, and each radio ad reaches 6700 potential customers. The company wants to maximize the number of ad exposures to potential customers.\r<BR>\n" );
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document.write( "Use  x for number of Newspaper advertisements and  y  for number of Radio advertisements. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852179 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52776)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=852179\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "A company has $14,830 available per month for advertising. Newspaper ads cost $190 each and can't run <BR>\n" );
document.write( "more than 24 times per month. Radio ads cost $590 each and can't run more than 32 times per month at this price.<BR>\n" );
document.write( "Each newspaper ad reaches 5700 potential customers, and each radio ad reaches 6700 potential customers. <BR>\n" );
document.write( "The company wants to maximize the number of ad exposures to potential customers.<BR>\n" );
document.write( "Determine the most profitable / (effective) way to do it.<BR>\n" );
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document.write( "To find the maximum number of ad exposures, let's formulate the problem in terms \r\n" );
document.write( "of objective function and constraints.  \r\n" );
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document.write( "Let x be the number of newspaper ads and y be the number of radio ads. \r\n" );
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document.write( "The objective function is \r\n" );
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document.write( "    P = 5700*x + 6700*y.       (1)\r\n" );
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document.write( "It is the number of possible expositions, and we want to maximize it.\r\n" );
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document.write( "The constraints are: \r\n" );
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document.write( "    190*x + 590*y ≤ 14830 (the budget),     (2)\r\n" );
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document.write( "    x ≤ 24 (newspaper ad limit), y ≤ 32 (radio ad limit),     (3)\r\n" );
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document.write( "    x ≥ 0, y ≥ 0 (non-negativity).     (4)\r\n" );
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document.write( "Now it is presented as a typical Linear Programming problem. But it can be easily solved MENTALLY\r\n" );
document.write( "using \"the most aggressive\" logical strategy/methodology.\r\n" );
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document.write( "From expression (1) for the objective function, we see that the contribution of each single newspaper ad\r\n" );
document.write( "(in terms of the number of potential expositions, 5700) is comparable with (or distinct insignificantly from) \r\n" );
document.write( "the contribution of each single radio ad (6700).\r\n" );
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document.write( "But each newspaper ad is much cheaper ($190) than each radio ad ($590). So, it is clear that \r\n" );
document.write( "the most profitable strategy is  to make as many newspaper ads as possible (x=24), and then\r\n" );
document.write( "to spend the rest of the budget for the radio ads. \r\n" );
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document.write( "Thus the most effective solution is to make 24 newspaper ads, spending 24*190 = 4560 dollars for it.\r\n" );
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document.write( "The rest of the budget is then  $14830 - $4560 = $10270.\r\n" );
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document.write( "This amount can be / (should be) spent for radio ads.\r\n" );
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document.write( "It provides the number of radio ads  y = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10270%2F590 ALIGN=MIDDLE  ALT=\"10270%2F590\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 17.40678,\r\n" );
document.write( "and we should round this decimal number to the closest lesser integer number, which is y = 17.\r\n" );
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document.write( "So, the answer to the problem's question is THIS:\r\n" );
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document.write( "    24 newspaper ads and 17 radio ads provide the greatest possible number of expositions (~ potential customers),\r\n" );
document.write( "    which is then  5700*24 + 6700*17 = 250700.\r\n" );
document.write( "</PRE>\r<BR>\n" );
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document.write( "Solved.\r<BR>\n" );
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document.write( "<H3>Post-solution note</H3>\r<BR>\n" );
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document.write( "In this concrete problem, the presented method has one important advantage comparing with the traditional form<BR>\n" );
document.write( "geometric solution of Linear Programming problems.\r<BR>\n" );
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document.write( "Working in the frame of traditional Linear Programming geometric method, you will get the solution with non-integer decimals, <BR>\n" );
document.write( "so, you will be forced to use other arguments to complete the traditional solution.\r<BR>\n" );
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document.write( "Working in the way, presented here in the solution above, you will get the answer in integer numbers without any complications.\r<BR>\n" );
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document.write( "In other words, this problem is for INTEGER Linear Programming - not for regular Linear Programming.<BR>\n" );
document.write( "But integer Linear Programming problems require their special solution methodology <BR>\n" );
document.write( "(which is not studied in the school Math) and requires special solvers.<BR>\n" );
document.write( "Or, as it is done in my solution above - a special logical treatment.\r<BR>\n" );
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