document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1198959>Question 1198959</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A factory makes two types of beds, type A and type B. Each month, a number a number of type A and a number of type B are produced. The following constraints control monthly production:<BR>\n" );
document.write( "No more than 50 beds of Type A and no more than 40 beds of type B can be made.<BR>\n" );
document.write( "At least 60 beds in all must be made.<BR>\n" );
document.write( "The maximum number of beds that can be produced is 80.\r<BR>\n" );
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document.write( "The profit on type A is Php300 and on type B is Php150. How many beds on both types must be produced to maximize the profit? What is the minimum profit? </SPAN>\n" );
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document.write( "The very good response from the other tutor solves the problem in detail, using the standard process shown in most references: determine the corners of the feasibility region and evaluate the objective function at each corner.<br><BR>\n" );
document.write( "In fact it is almost always NOT necessary to evaluate the objective function at every corner of the feasibility region.  You can determine where the maximum value of the objective function will be obtained by comparing the slopes of the constraint functions and the objective function.<br><BR>\n" );
document.write( "The slanted constraint boundary lines are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%2BB%3E=60 ALIGN=MIDDLE  ALT=\"A%2BB%3E=60\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%2BB%3C=80 ALIGN=MIDDLE  ALT=\"A%2BB%3C=80\"   DISABLED_style= \"max-width:90%; height:auto;\" >; both of those lines have a slope of -1.<br><BR>\n" );
document.write( "The objective function is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=300A%2B150B ALIGN=MIDDLE  ALT=\"300A%2B150B\"   DISABLED_style= \"max-width:90%; height:auto;\" >; that line has a slope of -2.<br><BR>\n" );
document.write( "The maximum (and minimum!) values of the objective function will occur where a line with slope -2 just touches the feasibility region.<br><BR>\n" );
document.write( "In this problem, a quick sketch showing the feasibility region will easily show that the maximum value of the objective function will occur where the constraint boundary lines x=50 and x+y=80 intersect, at (50,30).<br><BR>\n" );
document.write( "So the maximum profit is 300(50)+150(30) = 15000+4500 = 19500, when 50 beds of type A and 30 of type B are produced.<br><BR>\n" );
document.write( "ANSWERS: 50 type A, 30 type B; profit 19500<br><BR>\n" );
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