document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=460454>Question 460454</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I am really struggling with these homework problems and could use some assistance in figuring out how to solve them. \r<BR>\n" );
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document.write( "3. Suppose we are to maximize the objective function 4x + 3y over a feasible set that has vertices (1, 2), (5, 0), (8, 3), and (6, 4). The maximum objective function value is ______. I answered (8,3) but it was incorrect. How do I solve this?\r<BR>\n" );
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document.write( "4. The XYZ Corporation plans to open three different types of fast-food restaurants. Each type of A restaurant requires an initial cash outlay of $600,000, needs 15 employees, and is expected to make an annual profit of $45,000. Each type B restaurant requires an initial cash outlay of $400,000, needs 9 employees, and is expected to make an annual profit of $35,000. Each type C restaurant requires an initial cash outlay of $300,000, needs 5 employees, and is expected to make an annual profit of<BR>\n" );
document.write( "$25,000. The XYZ Corporation has $48,000,000 available for initial outlays, does not want to hire more than 1000 new employees, and would like to open at most 70 restaurants. How many restaurants<BR>\n" );
document.write( "of each type should the XYZ Corporation open in order to maximize the expected annual profit? Let x = number of type A restaurants, y = number of type B restaurants, and z = number of type C restaurants. \r<BR>\n" );
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document.write( "I said 45000x+35000y+25000z = 70 subject to <BR>\n" );
document.write( "{x+y+z<=45000<BR>\n" );
document.write( "600x+400y+300z<=35000<BR>\n" );
document.write( "15x+9y+5z<=25000<BR>\n" );
document.write( "x,y,z>=0\r<BR>\n" );
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document.write( "I got it wrong as well. Please show me how to do these two problems.<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 #315974 by </B><A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><B>Edwin McCravy(20056)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><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=Edwin+McCravy><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=315974\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 3. Suppose we are to maximize the objective function 4x + 3y over a feasible set that has vertices (1, 2), (5, 0), (8, 3), and (6, 4). The maximum objective function value is ______. I answered (8,3) but it was incorrect. How do I solve this?<BR>\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "You didn't read the question carefully.  You were asked\r\n" );
document.write( "to give the maximum value of 41, not the coordinates of\r\n" );
document.write( "the vertex at which  this maximum occurred.  You were right \r\n" );
document.write( "that (8,3) is the vertex at which the maximum value occurs,\r\n" );
document.write( "but that's not what the problem asked for. \r\n" );
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document.write( "When you evaluate 4x + 3y at each of the\r\n" );
document.write( "vertices:\r\n" );
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document.write( "x  y |        4x + 3y\r\n" );
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document.write( "1  2 | 4(1)+3(2) = 4+6 = 10\r\n" );
document.write( "5  0 | 4(5)+3(0) = 20+0 = 20\r\n" );
document.write( "8  3 | 4(8)+3(3) = 32+9 = 41\r\n" );
document.write( "6  4 | 4(6)+3(4) = 24+12 = 36\r\n" );
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document.write( "So the correct answer is 41, not (8,3)\r\n" );
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document.write( "</PRE><BR>\n" );
document.write( "4. The XYZ Corporation plans to open three different types of fast-food restaurants. Each type of A restaurant requires an initial cash outlay of $600,000, needs 15 employees, and is expected to make an annual profit of $45,000. Each type B restaurant requires an initial cash outlay of $400,000, needs 9 employees, and is expected to make an annual profit of $35,000. Each type C restaurant requires an initial cash outlay of $300,000, needs 5 employees, and is expected to make an annual profit of<BR>\n" );
document.write( "$25,000. The XYZ Corporation has $48,000,000 available for initial outlays, does not want to hire more than 1000 new employees, and would like to open at most 70 restaurants. How many restaurants<BR>\n" );
document.write( "of each type should the XYZ Corporation open in order to maximize the expected annual profit? Let x = number of type A restaurants, y = number of type B restaurants, and z = number of type C restaurants. <BR>\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "We get the cash outlay statements together:\r\n" );
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document.write( "Each type A restaurant requires an initial cash outlay of $600,000\r\n" );
document.write( "Each type B restaurant requires an initial cash outlay of $400,000\r\n" );
document.write( "Each type C restaurant requires an initial cash outlay of $300,000\r\n" );
document.write( "The XYZ Corporation has $48,000,000 available for initial outlays\r\n" );
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document.write( "Total of cash outlays = 600000x + 400000y + 300000z &#8806; 48000000\r\n" );
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document.write( "We get the employees statements together:\r\n" );
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document.write( "Each type A restaurant...needs 15 employees\r\n" );
document.write( "Each type B restaurant...needs 9 employees\r\n" );
document.write( "Each type C restaurant...needs 5 employees\r\n" );
document.write( "The XYZ Corporation...does not want to hire more than 1000 new employees\r\n" );
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document.write( "Total number of employees = 15x + 9y + 5z &#8806; 1000\r\n" );
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document.write( "We get the expected profit statements together:\r\n" );
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document.write( "Each type A restaurant...is expected to make an annual profit of $45,000\r\n" );
document.write( "Each type B restaurant...is expected to make an annual profit of $35,000.\r\n" );
document.write( "Each type C restaurant...is expected to make an annual profit of $25,000.\r\n" );
document.write( "Total expected profit = 45000x+35000y+25000z = P\r\n" );
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document.write( "This is what we are to maximize.  So the problem is to\r\n" );
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document.write( "maximize P = 45000x+35000y+25000z\r\n" );
document.write( "Subject to the constraints:\r\n" );
document.write( "600000x + 400000y + 300000z &#8806; 48000000\r\n" );
document.write( "              15x + 9y + 5z &#8806; 1000\r\n" );
document.write( "                      x,y,z &#8807; 0\r\n" );
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document.write( "That may be what you're looking for without solving.\r\n" );
document.write( "I'll start to solve it though.\r\n" );
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document.write( "To solve it we can simplify it by letting the profit be\r\n" );
document.write( "in thousands of dollars rather than in dollars, and we can\r\n" );
document.write( "divide the first inequality through by 100000\r\n" );
document.write( "Then our \r\n" );
document.write( "we have:\r\n" );
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document.write( "maximize P = 45x+35y+25z subject to the constraints:\r\n" );
document.write( "               6x + 4y + 3z &#8806; 480\r\n" );
document.write( "              15x + 9y + 5z &#8806; 1000\r\n" );
document.write( "                      x,y,z &#8807; 0\r\n" );
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document.write( "So we put in two slack variables and make the equations:\r\n" );
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document.write( "               6x + 4y + 3z + s1 = 480\r\n" );
document.write( "              15x + 9y + 5z + s2 = 1000\r\n" );
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document.write( "Then we get 0 on the right of objective function:\r\n" );
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document.write( "                               P = 45x + 35y + 25z\r\n" );
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document.write( "            -45x - 36y - 25z + P = 0\r\n" );
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document.write( "And we have this system of equations:\r\n" );
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document.write( "               6x +  4y +  3z + s1           =  480\r\n" );
document.write( "              15x +  9y +  5z      + s2      = 1000\r\n" );
document.write( "             -45x - 36y - 25z            + P =    0\r\n" );
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document.write( "Now we make this matrix:\r\n" );
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document.write( "           &#9150;  6   4   3 | 1  0 | 0 |  480&#9163;\r\n" );
document.write( "           &#9122; 15   9   5 | 0  1 | 0 | 1000&#9122;\r\n" );
document.write( "           &#9122;—————————————————————————————&#9122;\r\n" );
document.write( "           &#9151;-45 -36 -25 | 0  0 | 1 |    0&#9164;\r\n" );
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document.write( "Do you want me to finish the Simplex method here?\r\n" );
document.write( "It get's pretty messy.  Maybe you were just supposed\r\n" );
document.write( "to set it up.  If you were to solve it, post again\r\n" );
document.write( "or tell me in a thank-you note.\r\n" );
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document.write( "Edwin</PRE>\r<BR>\n" );
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