document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1141303>Question 1141303</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A food factory is making a beverage for a customer from mixing two different existing products A and B. The compositions of A and B and prices ($/L) are given as follows,<BR>\n" );
document.write( "Amount (L) in /100 L of A and B<BR>\n" );
document.write( "  Lime Orange Mango Cost ($/L)<BR>\n" );
document.write( "A  3     6     4     8<BR>\n" );
document.write( "B  8     4     6     7<BR>\n" );
document.write( "The customer requires that there must be at least 4.5 Litres (L) Orange and at least 5 Litres of Mango concentrate per 100 Litres of the beverage respectively, but no more than 6 Litres of Lime concentrate per 100 Litres of beverage. The customer needs at least 70 Litres of the beverage per week.\r<BR>\n" );
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document.write( "a) Explain why a linear programming model would be suitable for this case study.<BR>\n" );
document.write( "b) Formulate a Linear Programming (LP) model for the factory that minimises the total cost of producing the beverage while satisfying all constraints.<BR>\n" );
document.write( "c) Use the graphical method to find the optimal solution. Show the feasible region and the optimal solution on the graph.What is the minimal cost for the product?<BR>\n" );
document.write( "d)Is there a range for the cost ($) of A that can be changed without affecting the optimum solution obtained above? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #761880 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=761880\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> x = number of liters of product A.<BR>\n" );
document.write( "y = number of liters of product B.\r<BR>\n" );
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document.write( "make a chart as shown 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;\">                            product A          product B\r\n" );
document.write( "lime per 100 liters                  3                  8\r\n" );
document.write( "orange per 100 liters                6                  4\r\n" );
document.write( "mango per 100 liters                 4                  6\r\n" );
document.write( "cost per liter                       8                  7\r\n" );
document.write( "number of liters variable            x                  y              \r\n" );
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document.write( "your objective function is to minimize the cost.\r<BR>\n" );
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document.write( "the equation for that is cost = 8x + 7y\r<BR>\n" );
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document.write( "The customer requires that there must be at least 4.5 liters (L) Orange and at least 5 liters of Mango concentrate per 100 liters of the beverage respectively, but no more than 6 liters of Lime concentrate per 100 liters of beverage. The customer needs at least 70 liters of the beverage per week. \r<BR>\n" );
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document.write( "equation for at least 4.5 liters of orange is (6x + 4y)/100 >= 4.5\r<BR>\n" );
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document.write( "equation for at least 5 liters of mango is (4x + 6y)/100 >= 5\r<BR>\n" );
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document.write( "equation for not more than 6 liters of lime is (3x + 8y)/100 <= 6\r<BR>\n" );
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document.write( "equation for the customer needs at least 70 liters of the beverage every week is x + y >= 70\r<BR>\n" );
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document.write( "solve for y in each of these equations to get:\r<BR>\n" );
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document.write( "(6x + 4y)/100 >= 4.5 becomes y >= (450 - 6x) / 4\r<BR>\n" );
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document.write( "(4x + 6y)/100 >= 5 becomes y >= (500 - 4x) / 6\r<BR>\n" );
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document.write( "(3x + 8y)/100 <= 6 becomes y <= (600 - 3x) / 8\r<BR>\n" );
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document.write( "x + y >= 70 becomes y >= 70 - x\r<BR>\n" );
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document.write( "since the value of x and y can't be negative, then 2 more constraint equations are:\r<BR>\n" );
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document.write( "x >= 0<BR>\n" );
document.write( "y >= 0\r<BR>\n" );
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document.write( "using the desmos.com calculator, you would then graph the OPPOSITE of these inequalities.\r<BR>\n" );
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document.write( "the area of the graph that is NOT shaded is your region of feasibility.\r<BR>\n" );
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document.write( "the corner points of this feasible region would then contain the least cost solution.\r<BR>\n" );
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document.write( "your constraint equations are:\r<BR>\n" );
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document.write( "(6x + 4y)/100 >= 4.5<BR>\n" );
document.write( "(4x + 6y)/100 >= 5<BR>\n" );
document.write( "(3x + 8y)/100 <= 6<BR>\n" );
document.write( "x + y >= 70<BR>\n" );
document.write( "x >= 0<BR>\n" );
document.write( "y >= 0\r<BR>\n" );
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document.write( "the equations after you have solved for y become:\r<BR>\n" );
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document.write( "y >= (450 - 6x) / 4<BR>\n" );
document.write( "y >= (500 - 4x) / 6<BR>\n" );
document.write( "y <= (600 - 3x) / 8<BR>\n" );
document.write( "y >= 70 - x<BR>\n" );
document.write( "x >= 0<BR>\n" );
document.write( "y >= 0\r<BR>\n" );
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document.write( "x >= 0 remains the same because there is no y in that equation to be solved for.\r<BR>\n" );
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document.write( "you will graph the opposite of these inequalities using the desmos.com calculator.\r<BR>\n" );
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document.write( "the equations to be graphed are:\r<BR>\n" );
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document.write( "y <= (450 - 6x) / 4<BR>\n" );
document.write( "y <= (500 - 4x) / 6<BR>\n" );
document.write( "y >= (600 - 3x) / 8<BR>\n" );
document.write( "y <= 70 - x<BR>\n" );
document.write( "x <= 0<BR>\n" );
document.write( "y <= 0\r<BR>\n" );
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document.write( "your graph will look like this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2019/052112.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "first graph is a full view without showing the corner points of the feasible region.\r<BR>\n" );
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document.write( "second graph is a full view showing two of the corner points of the feasible region.\r<BR>\n" );
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document.write( "third graph is a closeup view showing the other two corner points of the feasible region.\r<BR>\n" );
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document.write( "fourth graph is a full view showing all of the corner points of the feasible region.\r<BR>\n" );
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document.write( "the corner points of your feasible region are:\r<BR>\n" );
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document.write( "(33.333,62.5)<BR>\n" );
document.write( "(35,60)<BR>\n" );
document.write( "(125,0)<BR>\n" );
document.write( "(200,0)\r<BR>\n" );
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document.write( "you evaluate the objective function at each of these corner points to get:\r<BR>\n" );
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document.write( "(33.333,62.5) yields a cost of 704.164<BR>\n" );
document.write( "(35,60) yields a cost of 700<BR>\n" );
document.write( "(125,0) yields a cost of 1000<BR>\n" );
document.write( "(200,0) yields a cost of 1600\r<BR>\n" );
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document.write( "the lowest cost solution is the corner point of (35,60).\r<BR>\n" );
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document.write( "you would then evaluate that corner point to see if all the constraints have been met.\r<BR>\n" );
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document.write( "at the corner point of (35,60), .....\r<BR>\n" );
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document.write( "(6x + 4y)/100 >= 4.5 becomes (6*35 + 4*60)/100 >= 4.5 which becomes 4.5 >= 4.5 which is true.\r<BR>\n" );
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document.write( "(4x + 6y)/100 >= 5 becomes (4*35 + 6*60)/100 >= 5 which becomes 5 >= 5 which is true.\r<BR>\n" );
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document.write( "(3x + 8y)/100 <= 6 becomes (3*35 + 8*60)/100 <= 6 which becomes 4.5 <= 6 which is true.\r<BR>\n" );
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document.write( "x + y >= 70 becomes 35 + 60 >= 70 which becomes 95 >= 70 which is true.\r<BR>\n" );
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document.write( "x >= 0 and y >= 0 are both also true.\r<BR>\n" );
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document.write( "all constraint have been satisfied so the least cost solution is (35,60) which means that you would use 35 liters of solution A and 60 liters of solution B.\r<BR>\n" );
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document.write( "to find what cost for product A would make the least cost solution change hands, you would look at the two solutions that are closest to each other and see what cost of A would make the second lowest cost solution the least cost solution.\r<BR>\n" );
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document.write( "the second lowest cost solution will becomes the least cost solution when the following equation is satisfied.\r<BR>\n" );
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document.write( "note that some rounding in the second most least cost solution was used.<BR>\n" );
document.write( "to make this as accurate as possible, that rounding needs to be removed.\r<BR>\n" );
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document.write( "the second least cost solution corner point on the graph is shown as (33.333,62.5).\r<BR>\n" );
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document.write( "that is really (100/3,62.5)\r<BR>\n" );
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document.write( "the second least cost is therefore really 100/3 * 8 + 62.5 * 7 = 704 and 1/6.\r<BR>\n" );
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document.write( "remove the mixed fraction and the second least cost is 4225/6.\r<BR>\n" );
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document.write( "so your least and next least cost equations and solutions are.\r<BR>\n" );
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document.write( "100/3 * 8 + 62.5 * 7 = 4225/6<BR>\n" );
document.write( "35 * 8 + 60 * 7 = 700\r<BR>\n" );
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document.write( "you will be looking for the cost of product A, so make that cost equal to x rather than 8 to get the following equaitons.\r<BR>\n" );
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document.write( "100/3 * x + 62.5 * 7 = ???<BR>\n" );
document.write( "35 * x + 60 * 7 = ???\r<BR>\n" );
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document.write( "the least cost solution will change hands when the second least cost solution becomes less than the least cost solution.\r<BR>\n" );
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document.write( "that will occur when 35 * x + 60 * 7 > 100/3 * x + 62.5 * 7\r<BR>\n" );
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document.write( "simplify that equation (it's really an inequality) to get:\r<BR>\n" );
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document.write( "35 * x + 420 > 100/3 * x + 437.5\r<BR>\n" );
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document.write( "multiply  both sides of that inequality by 3 to get:\r<BR>\n" );
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document.write( "105 * x + 1260 > 100 * x + 1312.5\r<BR>\n" );
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document.write( "subtract 100 * x from both sides of that inequality and subtract 1260 from both sides of that inequality to get:\r<BR>\n" );
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document.write( "105 * x - 100 * x > 1312.5 - 1260\r<BR>\n" );
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document.write( "simplify to get 5 * x > 52.5\r<BR>\n" );
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document.write( "solve for x to get x > 10.5\r<BR>\n" );
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document.write( "when the cost for product A is greater than 10.5, the least cost solution will change hands with the next least cost solution.\r<BR>\n" );
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document.write( "for example, when x = 11, the cost at each corner point becomes:\r<BR>\n" );
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document.write( "(100/3,62.5) yields a cost of 100/3 * 11 + 62.5 * 7 = 804 and 1/6.<BR>\n" );
document.write( "(35,60) yields a cost of 35 * 11 + 60 * 7 = 805<BR>\n" );
document.write( "(125,0) yields a cost of 125 * 11 + 60 * 0 = 1375<BR>\n" );
document.write( "(200,0) yields a cost of 200 * 11 + 60 * 0 = 2200\r<BR>\n" );
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document.write( "the lowest cost solution is now at (100/3,62.5) rather than at (35,60).\r<BR>\n" );
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document.write( "this can't be seen very easily graphically, but the following spreadsheet shows at what cost per liter for product A that the change over takes place.\r<BR>\n" );
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document.write( "one minimum cost is shown in red.<BR>\n" );
document.write( "two equal minimum costs are shown in blue.\r<BR>\n" );
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document.write( "you can see from the spreadsheet that the minimum cost is at (35, 60) when the cost per liter for product A is less than 10.5.\r<BR>\n" );
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document.write( "when the cost per liter for product A is equal 10.5, then the minimum cost is shared between (35,60) and (33 and 1/3, 62.5).\r<BR>\n" );
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document.write( "when the cost per liter for product A is greater than 10.5, then the minimum cost is at (33 and 1/3, 62.5).\r<BR>\n" );
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document.write( "why was this problem suitable for a linear programming model?\r<BR>\n" );
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document.write( "from my perspective, there was an objective function which either had to be minimized or maximized, and they were constraints that had to be satisfied.\r<BR>\n" );
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document.write( "those were the two key factors that indicated to me that this was probably a linear programming type of problem.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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