document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1202803>Question 1202803</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Sally Sethness assembles stereo equipment for resale in her shop. She offers two products, turntables and cassette players. She makes a profit of $10 on each turntable and $6 on each cassette. Both must go through two steps in her shop—assembly and bench checking. A turntable takes 12 hours to assemble and 4 hours to bench check. A cassette player takes 4 hours to assemble but 8 hours to bench check. Looking at this month's schedule, Sally sees that she has 60 assembly hours uncommitted and 40 hours of bench-checking time available. Use graphic linear programming to find her best combination of turntables and cassette players. What is the total profit on the combination you found? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #837915 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52873)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=837915\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "Sally Sethness assembles stereo equipment for resale in her shop. <BR>\n" );
document.write( "She offers two products, turntables and cassette players. <BR>\n" );
document.write( "She makes a profit of $10 on each turntable and $6 on each cassette. <BR>\n" );
document.write( "Both must go through two steps in her shop—assembly and bench checking. <BR>\n" );
document.write( "A turntable takes 12 hours to assemble and 4 hours to bench check. <BR>\n" );
document.write( "A cassette player takes 4 hours to assemble but 8 hours to bench check. <BR>\n" );
document.write( "Looking at this month's schedule, Sally sees that she has 60 assembly hours uncommitted <BR>\n" );
document.write( "and 40 hours of bench-checking time available. <BR>\n" );
document.write( "Use graphic linear programming to find her best combination of turntables and cassette players. <BR>\n" );
document.write( "What is the total profit on the combination you found?<BR>\n" );
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document.write( "Let X be the number of turntable  and Y be the number of cassette players.\r\n" );
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document.write( "We want maximize the profit function\r\n" );
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document.write( "    P(X,Y) = 10X + 6Y\r\n" );
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document.write( "under the following restrictions\r\n" );
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document.write( "    12X + 4Y <= 60      (assembly time)\r\n" );
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document.write( "     4X + 8Y <= 40      (bench check time)\r\n" );
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document.write( "     X >= 0,  Y >= 0\r\n" );
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document.write( "The plot of the feasibility domain is shown in the Figure below.\r\n" );
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document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+400%2C+400%2C+-2%2C+22%2C+-2%2C+22%2C%0D%0A++++++++++%2860-12x%29%2F4%2C++%2840-4x%29%2F8%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r\n" );
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document.write( "    Plots y = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2860-12x%29%2F4 ALIGN=MIDDLE  ALT=\"%2860-12x%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >  (red) and y = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2840-4x%29%2F8 ALIGN=MIDDLE  ALT=\"%2840-4x%29%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > (green)\r\n" );
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document.write( "It is a quadrilateral in QI with the vertices  at points P1 = (0,0), P2 = (0,5), P3 = (4,3), P4 = (5,0).\r\n" );
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document.write( "We apply the standard Linear Programming method in its geometric interpretation.\r\n" );
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document.write( "The solution is one of these 4 points, where the objective function (profit) has a maximum.\r\n" );
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document.write( "You calculate the values of the objective function  P(X,Y)  (profit)  at listed points\r\n" );
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document.write( "    P1:  P(0,0) = 10*0 + 6*0 =    0,\r\n" );
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document.write( "    P2:  P(0,4) = 10*0 + 6*5 =   30,\r\n" );
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document.write( "    P3:  P(4,3) = 10*4 + 6*3 =   58,\r\n" );
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document.write( "    P4:  P(6,0) = 10*5 + 6*0 =   50.\r\n" );
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document.write( "Comparing these values, you find the optimal point.\r\n" );
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document.write( "It is  P3:  (X,Y) = (4,3),  which means  4 turntables and 3 cassette players, providing maximum PROFIT of 58 dollars.\r\n" );
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document.write( "Solved.\r<BR>\n" );
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document.write( "If you want to see many other similar and different solved problems, &nbsp;look into the lesson\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=https://www.algebra.com/algebra/homework/word/misc/Solving-minimax-problems-by--the-Linear-Programming-method.lesson>Solving minimax problems by the Linear Programming method</A> \r<BR>\n" );
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document.write( "in this site.\r<BR>\n" );
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document.write( "In opposite to Calculus, which was invented about three hundred years ago, <BR>\n" );
document.write( "the Linear Programming method for solving linear min-max problems<BR>\n" );
document.write( "was invented about only 80 years ago, in 1940'.\r<BR>\n" );
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document.write( "It quickly became widely accepted and widely extended method <BR>\n" );
document.write( "of solving many important problems in Economics.\r<BR>\n" );
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document.write( "The size of solved problems quickly increased from tens to hundreds,<BR>\n" );
document.write( "thousands and more variables. The best mathematical minds took part<BR>\n" );
document.write( "in creating and developing this method.\r<BR>\n" );
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document.write( "So, now this method is studied in high schools, colleges, and universities.\r<BR>\n" );
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document.write( "There are several levels of learning/teaching this method.\r<BR>\n" );
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document.write( "First, the lowest/starting level is its geometric version, as I presented it<BR>\n" );
document.write( "in my post. At this level, students learn it in high school. <BR>\n" );
document.write( "Usually, only two unknown variables are considered, and the purpose of <BR>\n" );
document.write( "this learning is to make first steps: to get a first acquaintance <BR>\n" );
document.write( "with the method, to get its idea, and to get familiarity on how it works.\r<BR>\n" );
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document.write( "    The major idea of the LP-method is that linear function over a polyhedron\r\n" );
document.write( "    in multi-dimensional linear space gets its maximum-minimum values\r\n" );
document.write( "    at the corners/vertices of the polyhedron.\r\n" );
document.write( "    The idea is simple, but enormously productive.\r\n" );
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document.write( "The second level is advanced consideration of the 2D case with two unknowns<BR>\n" );
document.write( "and 3D case. At this level, it is appropriate to consider version of the method<BR>\n" );
document.write( "with slopes of the constraint boundary lines and the slope of the objective function.<BR>\n" );
document.write( "It is the level of teaching this method in colleges, where Math is not a major specialty.\r<BR>\n" );
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document.write( "Third level is teaching students to use existing standard calculators<BR>\n" );
document.write( "from the Internet, like Theo does it in his post.<BR>\n" );
document.write( "This third level can sometimes be combined with the first or the second level.<BR>\n" );
document.write( "At this level, the major goal is to teach students to setup problems.\r<BR>\n" );
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document.write( "Fourth level is studying this method in the matrix version, oriented<BR>\n" );
document.write( "for work with thousands unknowns. It is intended for those students who are <BR>\n" );
document.write( "going/targeted to be the upper level users and/or professional DEVELOPERS <BR>\n" );
document.write( "in this area. It is the level of renowned Math colleges and Universities,<BR>\n" );
document.write( "and the level of graduate students, working on their PhD degrees in the area.\r<BR>\n" );
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document.write( "In the pedagogic literature, these levels are strictly separated.<BR>\n" );
document.write( "There are, probably, tens of textbooks for the first level;<BR>\n" );
document.write( "there are textbooks for the second level, although not so many.<BR>\n" );
document.write( "On my bookshelf, I have three college Math textbooks teaching at this level,<BR>\n" );
document.write( "and I am sure that every student, who really needs and wants to know<BR>\n" );
document.write( "the subject at this level, has an ability to learn from such textbooks.<BR>\n" );
document.write( "Always there is a possibility to buy affordable used textbook for low price,<BR>\n" );
document.write( "since at this level nothing is changed in decades. <BR>\n" );
document.write( "(And, empirically, than older is a textbook, than it is better).\r<BR>\n" );
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document.write( "For the highest, fourth level, there are standard textbooks and monographs<BR>\n" );
document.write( "on simplex method, as well as Proceedings of numerous conferences.<BR>\n" );
document.write( "They are used by PhD students in their Computer Science courses or in <BR>\n" );
document.write( "their Computational Math courses. And at the level 4, the simplex method technically <BR>\n" );
document.write( "has nothing in common with the levels 1, 2 and 3, except of general idea.\r<BR>\n" );
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document.write( "So, regarding the LP-method or simplex method, there are enough educational sources <BR>\n" );
document.write( "for each level, and they are strictly/accurately separated by levels of complexity.\r<BR>\n" );
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document.write( "Next, when students come to this forum asking to enlighten them about the LP-method,<BR>\n" );
document.write( "such requests are 100% from beginner students, who need to get basic knowledge at level number one.\r<BR>\n" );
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document.write( "As soon as they know the subject at this level one, they will not come to this forum <BR>\n" );
document.write( "asking to enlighten them at level two: they will go further on their own legs, <BR>\n" );
document.write( "without spikes/toggles.\r<BR>\n" );
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document.write( "It is why I always present them the LP-method at the level one, as I did it in my post above.\r<BR>\n" );
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document.write( "Not because I am a bad or a limited person, but because the life is arranged this way, and not otherwise.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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