document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1206298>Question 1206298</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A box company makes small and large wooden boxes.  Small boxes require 0,8 square metres of wood, while large ones require 1,4 square	metres.  All boxes require 0,5 hours of labour, regardless of size.  Wood is limited to 42 square metres, and only 24 hours of labour are available.  Due to warehouse space limitations, no more than 20 large boxes can be made each day.  Also, demand by customers for small boxes is limited to a maximum of 30 boxes.  Each small box yields a profit of R42,00 and each large box earns only R14,00.\r<BR>\n" );
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document.write( "Formulate a linear programming model for the company as follows;\r<BR>\n" );
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document.write( "(1)	Clearly identify the two decision variables in respect of each product to be produced.	\r<BR>\n" );
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document.write( "		Let x1 = _____________________________________________________\r<BR>\n" );
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document.write( "		Let x2 = _____________________________________________________\r<BR>\n" );
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document.write( "PAGE 4\r<BR>\n" );
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document.write( "(2)	Write down the objective function that maximizes total profit.\r<BR>\n" );
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document.write( "		Maximize z = ________________________________________________	\r<BR>\n" );
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document.write( "(3)	Formulate the constraints in respect of the two resources used in producing the bowls and mugs, warehouse space limitation, and demand limitation.\r<BR>\n" );
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document.write( "		______________________________________________	m2 of wood\r<BR>\n" );
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document.write( "		______________________________________________	labour	hours\r<BR>\n" );
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document.write( "		______________________________________________	warehouse\r<BR>\n" );
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document.write( "		______________________________________________	demand\r<BR>\n" );
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document.write( "(4)	Indicate that all constraints are non-negative.\r<BR>\n" );
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document.write( "			x1 ≥ 0		or 	x1, x2 ≥ 0<BR>\n" );
document.write( "			x2 ≥ 0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #843635 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=843635\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> x = number of small boxes.<BR>\n" );
document.write( "y = number of large boxes.\r<BR>\n" );
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document.write( "constraint inequalities are:<BR>\n" );
document.write( "y <= 20<BR>\n" );
document.write( "x <= 30<BR>\n" );
document.write( ".8x + 1.4y <= 42<BR>\n" );
document.write( ".5x + .5y <= 24<BR>\n" );
document.write( "x >= 0]<BR>\n" );
document.write( "y >= 0<BR>\n" );
document.write( "objective function:<BR>\n" );
document.write( "profit = 42x + 14y\r<BR>\n" );
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document.write( "using the decmos.com calculator, you would graph the opposite of the inequalities.<BR>\n" );
document.write( "the feasible region is the area on the graph that is not shaded.<BR>\n" );
document.write( "you would evaluate the objective function at each corner point to find the maximum profit.<BR>\n" );
document.write( "since partial boxes are not allowed, you would round to the next lowest integer.\r<BR>\n" );
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document.write( "the profit at each corner point is shown below.\r<BR>\n" );
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document.write( "(0,20) = 840<BR>\n" );
document.write( "(17.5,20) = (17,20) = 994<BR>\n" );
document.write( "(30,12.857) = (30,12) = 1428<BR>\n" );
document.write( "(30,0) = 1260\r<BR>\n" );
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document.write( "the maximum profit is at (20,12) where profit is equal to 42 * 30 + 12 * 14 = 1428.\r<BR>\n" );
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document.write( "all constraints are satisfied, such as:<BR>\n" );
document.write( "wood constraint at .8 * 30 + 1.4 * 12 = 40.8 <= 42 is satisfied.<BR>\n" );
document.write( "labor constraint at .5 * 30 + .5 * 12 = 21 <= 24 is satisfied.<BR>\n" );
document.write( "0 <= x <= 30 is satisfied.<BR>\n" );
document.write( "0 <= y <= 20 is satisfied.\r<BR>\n" );
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document.write( "here's what the graph looks like:\r<BR>\n" );
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document.write( " </SPAN>\n" );
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