document.write( "Question 1171978: The Electrocomp Corporation manufactures two electrical products: air conditioners and large fans. The assembly process for each is similar in that both require a certain amount of wiring and drilling. Each air conditioner takes 6 hours of wiring and 1 hours of drilling. Each fan must go through 2 hours of wiring and 1 hours of drilling. During the next production period, 240 hours of wiring time are available and up to 77 hours of drilling time may be used. Each air conditioner sold yields a profit of $20. Each fan assembles may be sold for a $15 profit. Formulate and solve this LP production mix situation to find the best combination of air conditioners and fans that yields the highest profit. Use the corner point graph approach \n" ); document.write( "
Algebra.Com's Answer #797387 by ikleyn(52800)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "The Electrocomp Corporation manufactures two electrical products: air conditioners and large fans.
\n" ); document.write( "The assembly process for each is similar in that both require a certain amount of wiring and drilling.
\n" ); document.write( "Each air conditioner takes 6 hours of wiring and 1 hour of drilling.
\n" ); document.write( "Each fan must go through 2 hours of wiring and 1 hour of drilling.
\n" ); document.write( "During the next production period, 240 hours of wiring time are available and up to 77 hours of drilling time may be used.
\n" ); document.write( "Each air conditioner sold yields a profit of $20. Each fan assembles may be sold for a $15 profit.
\n" ); document.write( "Formulate and solve this LP production mix situation to find the best combination of air conditioners and fans
\n" ); document.write( "that yields the highest profit. Use the corner point graph approach
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\n" ); document.write( "\n" ); document.write( "            The solution by @Theo contains several technical errors and one conceptual error, \r
\n" ); document.write( "\n" ); document.write( "            which leads to nonsensical answer in non-integer numbers \r
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\n" ); document.write( "\n" ); document.write( "                                     21.5 air conditioners and 55.5 large fans.\r
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\n" ); document.write( "\n" ); document.write( "            Therefore, I came to bring a correct solution.\r
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document.write( "Let x be the number of air conditioners and let y be the number of fans.\r\n" );
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document.write( "Then the objective function to maximize is the profit  \r\n" );
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document.write( "    P(x,y) = 20x + 15y  dollars.\r\n" );
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document.write( "The constraints are \r\n" );
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document.write( "    6x + 2y <= 240    (hours of wiring)\r\n" );
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document.write( "     x +  y <=  77    (hours of drilling)\r\n" );
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document.write( "    x >= 0,  y >= 0.\r\n" );
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document.write( "The constraints form THIS feasibility domain in the first quadrant QI, shown in the Figure below:\r\n" );
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document.write( "    \"graph%28+330%2C+330%2C+-20%2C+60%2C+-20%2C+140%2C%0D%0A++++++++++120-3x%2C+77-x%0D%0A%29\"\r\n" );
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document.write( "    Plot y = \"%28240-6x%29%2F2\" (red line) and  y = 77-x (green line)\r\n" );
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document.write( "Feasibility domain is a quadrilateral in QI  below both red and green lines adjacent to coordinate axes.\r\n" );
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document.write( "Feasibility domain has 3 vertices:\r\n" );
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document.write( "    P1 = (0,77)        (y-intercept to green line);\r\n" );
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document.write( "    P2 = (21.5,55.5)   (intersection point of the red and green line);\r\n" );
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document.write( "    P3 = (40,0)        (x-intercept to red line).\r\n" );
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document.write( "You should calculate the value of the objective function P(x,y) = 20x + 15y  at these three points:\r\n" );
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document.write( "    at  P1:   P(x,y) = 20*0     + 15*77    = 1155;\r\n" );
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document.write( "    at  P2:   P(x,y) = 20*21.5  + 15*55.5  = 1262.5;\r\n" );
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document.write( "    at  P3:   P(x,y) = 20*40    + 15*0     = 800.\r\n" );
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document.write( "Comparing the values, you see that the maximum is achieved at the point  P2 = (21.5,55.5),  where x= 21.5, y= 55.5,  and is equal to 1262.5.\r\n" );
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\n" ); document.write( "\n" ); document.write( "                        But STOP for a minute (!)\r
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\n" ); document.write( "\n" ); document.write( "This answer  MAKES  no  SENSE,  since we get the  \"solution\"  in non-integer numbers.\r
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\n" ); document.write( "\n" ); document.write( "Why we got an absurdist solution ? - Because we,  actually,  missed one important constraint at the setup step: \r
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\n" ); document.write( "\n" ); document.write( "                 The final answer must be in integer numbers.\r
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\n" ); document.write( "\n" ); document.write( "So,  this problem is not a simplest traditional Linear Programming problem.\r
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\n" ); document.write( "\n" ); document.write( "It is,  actually,  an  integer Linear Programming problem,  which means that the solution should be found among integer numbers.\r
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\n" ); document.write( "\n" ); document.write( "Unfortunately,  the naive simplest algorithm,  which works and traditionally is used for  LP-problems,\r
\n" ); document.write( "\n" ); document.write( " DOES NOT work  for  integer LP-problems.\r
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\n" ); document.write( "\n" ); document.write( "Which is even worst,  I could not find in the  Internet free of charge robust online solver for such integer  LP-problems.\r
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\n" ); document.write( "\n" ); document.write( "Nevertheless,  there is the way to complete the solution even in these non-favorable circumstances.\r
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\n" ); document.write( "\n" ); document.write( "The way is to trace all appropriate candidate integer points  (x,y)  in the coordinate plane;  calculate the  Profit function in all these points
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\n" ); document.write( "\n" ); document.write( "It can be done using  Excel,  as I describe it below.\r
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document.write( "In Excel, I formed a spreadsheet (see below).\r\n" );
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document.write( "The first column lists integer x-coordinates from 1 to 40.\r\n" );
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document.write( "The next column lists integer values of y = \"%28240-6x%29%2F2\" = 120 - 3x  for x = 1, 2, 3, . . . , 21.\r\n" );
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document.write( "The third column lists integer values of y = 77 - x  for x = 22, 23, 24, . . . , 40.\r\n" );
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document.write( "The last column is the profit function, calculated for these x- and y- integer values.\r\n" );
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document.write( "x	y = 77-x	y = 120-3x	P  = 20x+15y		\r\n" );
document.write( "1	76				1160		\r\n" );
document.write( "2	75				1165		\r\n" );
document.write( "3	74				1170		\r\n" );
document.write( "4	73				1175		\r\n" );
document.write( "5	72				1180		\r\n" );
document.write( "6	71				1185		\r\n" );
document.write( "7	70				1190		\r\n" );
document.write( "8	69				1195		\r\n" );
document.write( "9	68				1200		\r\n" );
document.write( "10	67				1205		\r\n" );
document.write( "11	66				1210		\r\n" );
document.write( "12	65				1215		\r\n" );
document.write( "13	64				1220		\r\n" );
document.write( "14	63				1225		 \r\n" );
document.write( "15	62				1230		\r\n" );
document.write( "16	61				1235		\r\n" );
document.write( "17	60				1240		\r\n" );
document.write( "18	59				1245		\r\n" );
document.write( "19	58				1250		\r\n" );
document.write( "20	57				1255		\r\n" );
document.write( "21	56				1260	<<<---===	\r\n" );
document.write( "22			54		1250		\r\n" );
document.write( "23			51		1225		\r\n" );
document.write( "24			48		1200		\r\n" );
document.write( "25			45		1175		\r\n" );
document.write( "26			42		1150		\r\n" );
document.write( "27			39		1125		\r\n" );
document.write( "28			36		1100		\r\n" );
document.write( "29			33		1075		\r\n" );
document.write( "30			30		1050		\r\n" );
document.write( "31			27		1025		\r\n" );
document.write( "32			24		1000		\r\n" );
document.write( "33			21		975		\r\n" );
document.write( "34			18		950		\r\n" );
document.write( "35			15		925		\r\n" );
document.write( "36			12		900		\r\n" );
document.write( "37			9		875		\r\n" );
document.write( "38			6		850		\r\n" );
document.write( "39			3		825		\r\n" );
document.write( "40			0		800		\r\n" );
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document.write( "The maximum profit is 1260 dollars;  it corresponds to the values of x= 21  and  y= 56.\r\n" );
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document.write( "Thus the problem is just solved, and the answer is:  21 air conditioner and 56 large fans provide the maximum profit of 1260 dollars.\r\n" );
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