document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1202927>Question 1202927</A>: <I><SPAN STYLE=\"word-break: break-all;\"> CalJuice Company has decided to introduce three fruit juices made from blending two or more concentrates. These juices will be packaged in 2-qt (64-oz) cartons. One carton of pineapple-orange juice requires 8 oz each of pineapple and orange juice concentrates. One carton of orange-banana juice requires 12 oz of orange juice concentrate and 4 oz of banana pulp concentrate. Finally, one carton of pineapple-orange-banana juice requires 4 oz of pineapple juice concentrate, 8 oz of orange juice concentrate, and 4 oz of banana pulp concentrate. The company has decided to allot 16,000 oz of pineapple juice concentrate, 24,000 oz of orange juice concentrate, and 5000 oz of banana pulp concentrate for the initial production run. The company also stipulated that the production of pineapple-orange-banana juice should not exceed 800 cartons. Its profit on one carton of pineapple-orange juice is $1.00, its profit on one carton of orange-banana juice is $0.80, and its profit on one carton of pineapple-orange-banana juice is $0.90. To realize a maximum profit, how many cartons of each blend should the company produce?<BR>\n" );
document.write( "pineapple-orange juice ____   cartons<BR>\n" );
document.write( "orange-banana juice  ___ cartons<BR>\n" );
document.write( "pineapple-orange-banana juice   800 cartons\r<BR>\n" );
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document.write( "What is the largest profit it can realize?<BR>\n" );
document.write( "$ ______<BR>\n" );
document.write( " \r<BR>\n" );
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document.write( "Are there any concentrates left over? (If so, enter the amount remaining. If not, enter 0.)<BR>\n" );
document.write( "pineapple concentrate    	<BR>\n" );
document.write( "0 oz<BR>\n" );
document.write( "orange concentrate    	<BR>\n" );
document.write( "0 oz<BR>\n" );
document.write( "banana pulp concentrate    	<BR>\n" );
document.write( " ____ oz<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852259 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52799)</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=852259\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> . 1202927<BR>\n" );
document.write( "CalJuice Company has decided to introduce three fruit juices made from blending two or more concentrates. <BR>\n" );
document.write( "These juices will be packaged in 2-qt (64-oz) cartons. <BR>\n" );
document.write( "One carton of pineapple-orange juice requires 8 oz each of pineapple and orange juice concentrates. <BR>\n" );
document.write( "One carton of orange-banana juice requires 12 oz of orange juice concentrate and 4 oz of banana pulp concentrate. <BR>\n" );
document.write( "Finally, one carton of pineapple-orange-banana juice requires 4 oz of pineapple juice concentrate, <BR>\n" );
document.write( "8 oz of orange juice concentrate, and 4 oz of banana pulp concentrate. <BR>\n" );
document.write( "The company has decided to allot 16,000 oz of pineapple juice concentrate, 24,000 oz of orange juice concentrate, <BR>\n" );
document.write( "and 5000 oz of banana pulp concentrate for the initial production run. <BR>\n" );
document.write( "The company also stipulated that the production of pineapple-orange-banana juice should not exceed 800 cartons. <BR>\n" );
document.write( "Its profit on one carton of pineapple-orange juice is $1.00, its profit on one carton of orange-banana juice is $0.80, <BR>\n" );
document.write( "and its profit on one carton of pineapple-orange-banana juice is $0.90. <BR>\n" );
document.write( "To realize a maximum profit, how many cartons of each blend should the company produce?<BR>\n" );
document.write( "pineapple-orange juice ____ cartons<BR>\n" );
document.write( "orange-banana juice ___ cartons<BR>\n" );
document.write( "pineapple-orange-banana juice 800 cartons<BR>\n" );
document.write( "What is the largest profit it can realize?<BR>\n" );
document.write( "$ ______\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Are there any concentrates left over? (If so, enter the amount remaining. If not, enter 0.)<BR>\n" );
document.write( "pineapple concentrate<BR>\n" );
document.write( "0 oz<BR>\n" );
document.write( "orange concentrate<BR>\n" );
document.write( "0 oz<BR>\n" );
document.write( "banana pulp concentrate<BR>\n" );
document.write( "____ oz<BR>\n" );
document.write( "~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\r<BR>\n" );
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document.write( "The solution to this problem in the post by @Theo is incorrect.\r<BR>\n" );
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document.write( "Indeed, his solution states\r<BR>\n" );
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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( "    the maximum profit is made when 800 cartons of type A and 1250 cartons of type C mixture are made and sold.\r\n" );
document.write( "    no cartons of type B mixture were made and sold.\r\n" );
document.write( "    Then the maximum profit is 800*1 + 1250*0.9 = 1925 dollars.\r\n" );
document.write( "</PRE>\r<BR>\n" );
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document.write( "I will disprove @Theo' solution right now. \r<BR>\n" );
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document.write( "For it, I will present another possible solution, which provides greater profit <BR>\n" );
document.write( "and still does satisfy the restrictions.\r<BR>\n" );
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document.write( "This solution is 2000 cartoons of pineapple-orange juice and 0 cartons for two other juice.\r<BR>\n" );
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document.write( "It requires 2000*8 = 16000 oz of pineapple concentrate (which is under restriction) <BR>\n" );
document.write( "        and 2000*8 = 16000 oz of orange concentrate    (which is also under restriction).\r<BR>\n" );
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document.write( "But the profit in this case is 1*2000 = 2000 dollars, which is GREATER than 1925 dollars in the @Theo' solution.\r<BR>\n" );
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document.write( "Thus I disproved the @Theo solution - - - it does not provide the optimum strategy.<BR>\n" );
document.write( "</pre>\r<BR>\n" );
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document.write( "Now, as I proved that the solution in the post by @Theo is incorrect,<BR>\n" );
document.write( "I will present my analysis below.\r<BR>\n" );
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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( "Let x be the number of cartons of pineapple-orange        PO-juice;\r\n" );
document.write( "    y be the number of cartons of orange-banana           OB-juice;\r\n" );
document.write( "    z be the number of cartons of pineapple-orange-banana POB-juice.\r\n" );
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document.write( "One carton PO-juice  = (8 oz P,  8 oz O        ).\r\n" );
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document.write( "One carton OB-juice  = (        12 oz O, 4 oz B).\r\n" );
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document.write( "One carton POB-juice = (4 oz P,  8 oz O, 4 oz B). \r\n" );
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document.write( "We want to maximize the profit function P = x + 0.8y + 0.9z dollars.\r\n" );
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document.write( "The restrictions are\r\n" );
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document.write( "    8x +       4z <= 16000  (pineapple juice concentrate restriction, in oz);\r\n" );
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document.write( "    8x + 12y + 8z <= 24000  (orange    juice concentrate restriction, in oz);\r\n" );
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document.write( "          4y + 4z <= 5000   (banana    pulp  concentrate,             in oz).\r\n" );
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document.write( "    z <= 800,\r\n" );
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document.write( "    x >= 0,  y >= 0,  z >= 0.\r\n" );
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document.write( "I will solve the problem using the simplex method and the online solver at this web-site\r\n" );
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document.write( "https://www.zweigmedia.com/RealWorld/simplex.html\r\n" );
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document.write( "(the same solver, as @Theo used).\r\n" );
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document.write( "It requires the input information be presented in its special format, which you may find \r\n" );
document.write( "in the description to the solver.\r\n" );
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document.write( "So, the formatted input information for the solver is this\r\n" );
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document.write( "maximize p = x + 0.8y + 0.9z subject to\r\n" );
document.write( "8x + 4x <= 16000\r\n" );
document.write( "8x + 12y + 8z <= 24000\r\n" );
document.write( "4y + 4z <= 5000\r\n" );
document.write( "z <= 800,\r\n" );
document.write( "x >= 0,\r\n" );
document.write( "y >= 0,\r\n" );
document.write( "z >= 0.\r\n" );
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document.write( "I copy-paste it into the solver.\r\n" );
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document.write( "After one click, the solver generates this OPTIMAL SOLUTION\r\n" );
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document.write( "    p = 2640; x = 1600, y = 400, z = 800.\r\n" );
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document.write( "Having it, you can answer all remaining questions.\r\n" );
document.write( "</PRE>\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );