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 #639199 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=639199\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> you did pretty good in setting up the problem.\r<BR>\n" );
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document.write( "x + y = 5 should be x + y <= 5, since it is possible that the constraints can be satisfied with less than 5 driver.\r<BR>\n" );
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document.write( "15x + 10y = 50 can also be3 15x + 10y >= 50 since there is no restriction that says exactly 50 olympians need to be carried.\r<BR>\n" );
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document.write( "x has to be >= 0 and y has to be >= 0.\r<BR>\n" );
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document.write( "these constraints are usually there but not always specified.\r<BR>\n" );
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document.write( "the constraint equations that you need to graph are:\r<BR>\n" );
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document.write( "x <= 3<BR>\n" );
document.write( "y <= 4<BR>\n" );
document.write( "x + y <= 5 which is modeled as y <= 5 - x.<BR>\n" );
document.write( "15x + 10y >= 50 which is modeled as y >= (50 - 15x) / 10<BR>\n" );
document.write( "x >= 0<BR>\n" );
document.write( "y >= 0\r<BR>\n" );
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document.write( "you graph the equality portion and then you shade the inequality portion as required.\r<BR>\n" );
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document.write( "your graphed equations would look like this.\r<BR>\n" );
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document.write( "the shaded area represents your region of feasibility.\r<BR>\n" );
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document.write( "it is on or above the line x = 0 and on or to the right of the line y = 0.\r<BR>\n" );
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document.write( "it is on or to the left of the line x = 3 and on or below the line y = 4.\r<BR>\n" );
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document.write( "it is on or below the line y = 5 - x.\r<BR>\n" );
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document.write( "it is on or above the line y = (50 - 15x) / 10.\r<BR>\n" );
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document.write( "the corner points of the region of feasibility are:\r<BR>\n" );
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document.write( "intersection of the line y = (50 - 15x) / 10 and y = 4 = (2/3,4).\r<BR>\n" );
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document.write( "intersection of the line y = 5-x and y = 4 = (1,4).\r<BR>\n" );
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document.write( "intersection of the line x = 3 and y = 5-x = (3,2).\r<BR>\n" );
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document.write( "intersection of the line x = 3 and y = (50 - 15x) / 10 = (3,1/2).\r<BR>\n" );
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document.write( "as can be seen on the graph, those are the corner points of your region of feasibility.\r<BR>\n" );
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document.write( "now that you've found the region of feasbility, all that is left to do is to analyze the objective function at those intersection points.\r<BR>\n" );
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document.write( "your objective function should be maximum or minimum at those intersection points.\r<BR>\n" );
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document.write( "the constraint equations also need to be satisfied at those corner points if you graphed correctly.\r<BR>\n" );
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document.write( "the added wrinkle is that not all of the corner points are integers, and your solution needs to be integers.\r<BR>\n" );
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document.write( "you can't rent a fraction of a bus.\r<BR>\n" );
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document.write( "let's look at the points in turn.\r<BR>\n" );
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document.write( "the points are (2/3,4), (1,4), (3,2), (3,1/2).\r<BR>\n" );
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document.write( "the point (2/3,4) is not possible.<BR>\n" );
document.write( "it either has to be (0,4) or (1,4).<BR>\n" );
document.write( "(0,4) won't do because then you can only carry 40 olympians.<BR>\n" );
document.write( "the point would have to be (1,4).<BR>\n" );
document.write( "(1,4) can carry 15 + 40 = 55 olympians.<BR>\n" );
document.write( "that's more than enough so the constraints of carrying are satisfied.<BR>\n" );
document.write( "it also satisfies all the other constraints since you have 5 or less drivers x is between 0 and 3, and y is between 0 and 4.<BR>\n" );
document.write( "so (1,4) is a valid intersection point.\r<BR>\n" );
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document.write( "(1,4) also happens to be another intersection point on the graph, so we'll just use that and forget about (2/3,4).\r<BR>\n" );
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document.write( "(3,2) is a valid point on the graph.<BR>\n" );
document.write( "it carries 3 * 15 + 2 * 10 = 65 olympians.<BR>\n" );
document.write( "that satisfies the requirement that 50 or more olympians be carried.<BR>\n" );
document.write( "the point doesn't make much sense since it's more type B buses than needed, but it is a valid point because it does satisfy all the requirements.<BR>\n" );
document.write( "x is between 0 and 3 and y is between 0 and 4 and it is on or below the line y = 5-x and it is on or above the line y = (50-15x)/10.<BR>\n" );
document.write( "so it's another valid point that satisfies all the constraints.\r<BR>\n" );
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document.write( "the last point is (3,1/2).<BR>\n" );
document.write( "since you can't have 1/2 a bus, then we can round that up or down.<BR>\n" );
document.write( "down doesn't make sense since only 45 olympians can be carried.<BR>\n" );
document.write( "1 makes more sense, so we'll go with (3,1).\r<BR>\n" );
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document.write( "(3,1) satisfies all the constraints so it's a valid point.\r<BR>\n" );
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document.write( "we have 3 valid points of intersection, otherwise known as the corner points of the feabibility region.\r<BR>\n" );
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document.write( "they are (1,4), (3,2), (3,1).\r<BR>\n" );
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document.write( "what's left is to evaluate the objective function at these corner points.\r<BR>\n" );
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document.write( "at (1,4), the cost is 100 * 1 + 80 * 4 = 420<BR>\n" );
document.write( "at (3,2), the cost is 300 + 160 = 560.<BR>\n" );
document.write( "at (3,1), the cost is 300 + 80 = 380.\r<BR>\n" );
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document.write( "the minimum cost is at (3,1).\r<BR>\n" );
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document.write( "that's when they rent 3 type A buses and 1 type B bus.\r<BR>\n" );
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document.write( "they will carry 55 olympians which is greater than or equal to 50.\r<BR>\n" );
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document.write( "you can pick any other point within the feasibility region and it will not be less than 380 in cost.\r<BR>\n" );
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document.write( "at least that's the theory.\r<BR>\n" );
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document.write( "unfortunately, the point (2,2) is in the feasibility region, but it is not one of the corner points.\r<BR>\n" );
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document.write( "the cost would be 2 * 100 + 2 * 80 = 360.<BR>\n" );
document.write( "50 olympians can be carried.<BR>\n" );
document.write( "only 4 drivers are required.\r<BR>\n" );
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document.write( "why this was not one of the corner points is beyond me.\r<BR>\n" );
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document.write( "it is on the line y = (50 - 15x) / 10, however, and it does satisfy all the other constraint, so it is definitely in the feasibility region.\r<BR>\n" );
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document.write( "it does appear to be the minimum cost solution even though it's not on one of the corner points.\r<BR>\n" );
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document.write( "this is something you might want to bring up with your instructor.\r<BR>\n" );
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document.write( "i can't explain it.\r<BR>\n" );
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document.write( "the method that i showed you is correct, however, despite that wrinkle.\r<BR>\n" );
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document.write( "the only change i would make is to have 15x + 10y = 50 rather than 15x + 10y >= 50.\r<BR>\n" );
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document.write( "this would force the solution to be on the line.\r<BR>\n" );
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document.write( "the feasibility region would have to be on the line only.\r<BR>\n" );
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document.write( "the region of feasibility would just be on the line.\r<BR>\n" );
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document.write( "the point (2,2) is still not on a corner of the feasibility region since the only corner points would now be (2/3,4) and (3,1/2).\r<BR>\n" );
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document.write( "bottom line is i have no idea what the instructor was trying to show you with this problem.\r<BR>\n" );
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document.write( "talk to him / her about it.\r<BR>\n" );
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document.write( "see what he or she says.\r<BR>\n" );
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document.write( "let me know what the answer is if you do get an answer.\r<BR>\n" );
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document.write( "this problem is not the usual type of problem.\r<BR>\n" );
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document.write( "most of the time they work out and the solution is at the corner points.\r<BR>\n" );
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document.write( "it doesn't appear to be so with this problem.\r<BR>\n" );
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document.write( "i don't see anything in here that would make (2,2) a corner point, even though (2,2) is cheaper (360 versus 380) and it does carry 50 olympians (2*15 + 2*10 = 50), and it does require 5 or less drivers (4 actually).\r<BR>\n" );
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document.write( "it's possible i did something wrong, but i'll be damned if i can see it, and i have done many of these in the past successfully.\r<BR>\n" );
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