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##### MATH2070 Assignment Problems

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Question;Math2070 and Math2970;Optimisation and;Financial Mathematics;Assignment Problems;(term 2, year 2014);MATH2070: Do Questions 1{4;(not 5). Formulate Questions 1{3 as LP problems. Solve Question 1 graphically;and Questions 2{4 using the simplex algorithm. Do Question 2 by hand. Use;the MATLAB commands from the labs (not the linprog command) to do;Questions 3 and 4. The computer printout of your computing solutions must;contain the banner indicating your username. Any errors and blank lines should;be edited out. You have the option to use the MATLAB commands from the labs;(not the linprog command) to do Questions 2, 3;and 4. If typing your solution: copy the tableaus that Matlab outputs and paste;them into your report, and be sure to type full sentences explaining each step;(except for question #4 which doesn't require explanation). If;writing/computing your solutions by hand, be sure to write explanations for #2;and #3. Submit your folder (wallet style preferred) in the correct box opposite;the lifts on Carslaw Level 6.;MATH2970: Do Questions 1{3;5 (not 4). Follow the same instructions.;1. A tea company sells tea;under a \blue ribbon label" and an \economy label". Both are blended;from three basic grades of tea;Blue Ribbon;=;50% A;+;30% B;+;20% C;Economy;=;25% A;+;30% B;+;45% C;The;market prices are $800/tonne for blue ribbon and $650/tonne for economy. One;week the rm is given the option of buying up to 110 tonnes of grade A at;$700/tonne, 130 tonnes of grade B at $600/tonne and 220 tonnes of grade C at;$400/tonne. Calculate the pro t per tonne of each blend of tea. How much of;each blend should the company produce to maximize its pro t (not gross but net;returns) and what is the maximum pro t?;2.;A small manufacturing;company produces three products, A, B and C at its plant. The parts of each;product must be manufactured, the products are then assembled and nally the;nished products packaged;for;distribution to wholesalers. Each unit of product A takes 5 hours to;manufacture, 2 hours to assemble and 1 hour to package. The corresponding times;for product B are 212 hours, 1 hour and 40;minutes, and for product C, 412 hours, 112 hours and 40 minutes. The;manufacturer has up to 300 hours of manufacturing time per week using skilled;labour, 110 hours of assembling time, which uses semi-skilled labour, and 60;hours of packaging time which uses unskilled labour. If the pro ts per unit A;B and C are $110, $60 and $90, respectively, how many units of each product;should the manufacturer produce, assuming all product can be sold?;3.;A farming company owns two;farms, which di er in the growing of crops and their yields. Each farm has 100;acres available for cropping and 11,000 bushels of wheat and 5000 bushels of;corn must be grown. Farm A yields 400 bushels of wheat per acre at a cost of;$95 per acre and 500 bushels of corn per acre at a cost of $100 per acre. Farm;B yields 350 bushels of wheat per acre at a cost of $85 per acre and 650;bushels of corn per acre at a cost of $120 per acre. How should the crops be;planted to minimise the cost?;1;4. (MATH2070 only.) Formulate;the dual problem of the primal in Question 1. Solve the dual problem of;Question 1 and compare your answer with the primal problem.;5. (MATH2970 only.) A best;approximate solution to an inconsistent set of m equations in n unknowns;n;X;aij;xj = bi;i;= 1,:::, m;j=1;can be found by minimising the sum of;the absolute errors;m n;X;X;X;bi aij;xj;i=1 j=1;with respect to xk, k = 1,:::, n. This L1-approximation optimisation;problem is equivalent to the following LP problem with n + m decision variables;xj, j = 1,:::, n, and ei, i = 1,:::, m.;m;Minimise;z = ej;=1;n;Xj;subject;toei +;aij xj bi;i = 1,:::, m;=1;Xj;n;eiaij xjbi;i = 1,:::, m;=1;Xj;(a) Write down the dual problem;using dual variables yi, i = 1,:::, m, for the;rst m constraints and wi, i = 1,:::, m, for the;last m constraints. Show by eliminating the wi that the dual problem can;be simpli ed to;m;Maximise;v;=;biyi;=1;m;Xi;1;m;subject;to;aijyi=;aij,j = 1,:::, n;2;=1;i=1;Xi;X;0 yi 1;i = 1,:::, m;(b) Use the result of Part(a);to t a straight line of the form y = ax;+ b to the six points data points in;the;(x, y)-plane given by (2, 2), (1, 2), (1, 5), (2, 14), (3, 10), (5, 15) by;minimising the sum of the absolute errors at the six points (i.e. use L1 curve- tting). Formulate;the simpli ed dual LP problem of Part (a) for this curve- tting problem and;solve it using the MATLAB command linprog. (The decision variables m and b of;the primal problem are given by -lambda.eqlin | use the command in the form;[x,fval,exitflag,output,lambda] =;linprog(f,[],[],Aeq,beq,lb,ub,...;[],optimset('Display','iter','LargeScale','off','Simplex','on'));For further details on using the;command use help linprog in MATLAB or go to the website;http://www.mathworks.com.au/help/optim/ug/linprog.html;2;(c) Using the same data;points from part (b), t another line;that minimises the sum of squared errors;m n 2;X;X;X;bi aij;xj;i=1 j=1;Use Matlab's backslash command n with;help le here;http://www.mathworks.com.au/help/matlab/ref/mldivide.html;Plot;the data points and both of the straight lines you obtained in the same gure.;Which point is the outlier? Which line is less e ected by the outlier, and;hence a better t? Use a plot to show what happens to the lines if you add 25 to;the y-component of the outlier? Which tting method is robust to the outlier?;3

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