document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1023709>Question 1023709</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi, Would you please be able to help me. I need to understand the formula to add into a graphing tool for the following problem: <BR>\n" );
document.write( "A minibus operator is contracted to transport 50 Olympians from the Olympic Village to the Athletic Stadium.  He has three type A minibuses and four type B minibuses available. A type A minibus carries 15 people and a type B minibus carries 10 people.  Only five drivers are available.  It costs $100 to operate a type A minibus and $80 to operate a type B minibus.  He wishes to minimise the costs involved to transport the 50 Olympians.<BR>\n" );
document.write( "I've completed the following:<BR>\n" );
document.write( "Variables: Let x represent type A mini bus; Let y represent type B mini bus; Let c represent the minimum cost. <BR>\n" );
document.write( "Constraints: <BR>\n" );
document.write( "x <= 3 (At most 3 type A mini buses available)<BR>\n" );
document.write( "y <= 4 (At most 4 type B mini buses available)<BR>\n" );
document.write( "x + y = 5 (Limit on number of drivers)<BR>\n" );
document.write( "15x + 10y = 50 (Limit on number of passengers who can be carried in both mini buses)<BR>\n" );
document.write( "Objective Function: C = 100x + 80y<BR>\n" );
document.write( "Graphs - I've set the Axis to x: -1 to 10 and y: -1 to 10 and entered the above relations and constraints but don't know how to enter the formula to determine the minimum cost and I don't understand how corner points work. <BR>\n" );
document.write( "Thank you so much for any help you can offer. <BR>\n" );
document.write( "Chris\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #639460 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=639460\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "The general idea is to graph all of the inequalities and the area where they ALL overlap is called the \"feasibility area\".  Any ordered pair in the feasibility area will satisfy ALL of the constraints, and, if there is an optimum solution, it will be one of the feasibility area polygon vertices (what you called corner points).\r<BR>\n" );
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document.write( "The problem with graphing all of the inequalities as they are written is that it gets difficult to tell exactly where they all overlap.  To mitigate this difficulty I graph the inequalities in the OPPOSITE sense which results in the feasibility polygon having no shading; therefore much easier to see.\r<BR>\n" );
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document.write( "The constraint inequalities that you have are all correct except for two of them.  Yes, it is indeed true that you are contracting to haul 50 people, but this constraint needs to be couched as an inequality that says you have contracted to haul AT LEAST 50 passengers.  Also, since the MOST drivers you can have is 5, you need <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\ +\ y\ \leq\ 5\">\r<BR>\n" );
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document.write( "You also need to constrain both <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\"> to the positive integers.  You can't have a negative number of either type of bus, nor can you have a fractional number of buses of either type.  Hence, your constraints are:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x\ \leq\ 3\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ \leq\ 4\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x\ \in\ \mathbb{Z}\ :\ x\ \geq\ 0\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ \in\ \mathbb{Z}\ :\ y\ \geq\ 0\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x\ +\ y\ \leq\ 5\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 15x\ +\ 10y\ \geq\ 50\">\r<BR>\n" );
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document.write( "Now, when you go to graph these constraints, just flip the inequality symbol over.  I graphed all but the last constraint in the first diagram and then added the last one in the other figure.\r<BR>\n" );
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document.write( "Note the white quadrilateral that is formed.  This is your feasibility polygon.  Any points wholly within or on the boundaries of this polygon that have INTEGER coefficients are feasible solutions.  Constraining the solutions to the integers changes the rules about where to find the optimum solution.  In this case, you actually have four feasible points.\r<BR>\n" );
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document.write( "You did state the Objective Function correctly.  All you need to do is substitute the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\"> values of each of the four points into the objective function and do the arithmetic.  Since your objective is to minimize your objective function value, choose the point that gives you the smallest value of the objective function.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it\r<BR>\n" );
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