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##### Math 2070 and Math 2970 Term 2, 2014 Assignment Problems

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Question;Math2070 and Math2970Optimisation 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 graphicallyand Questions 2?4 using the simplex algorithm. Do Question 2 by hand. Use the MATLAB commands from thelabs (not the linprog command) to do Questions 3 and 4. The computer printout of your computing solutionsmust contain the banner indicating your username. Any errors and blank lines should be edited out. You havethe option to use the MATLAB commands from the labs (not the linprog command) to do Questions 2, 3and 4. If typing your solution: copy the tableaus that Matlab outputs and paste them into your report, and besure to type full sentences explaining each step (except for question #4 which doesn?t require explanation). Ifwriting/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 fromthree basic grades of tea:Blue Ribbon = 50% A + 30% B + 20% CEconomy = 25% A + 30% B + 45% CThe market prices are $800/tonne for blue ribbon and $650/tonne for economy. One week the firm is giventhe option of buying up to 110 tonnes of grade A at $700/tonne, 130 tonnes of grade B at $600/tonneand 220 tonnes of grade C at $400/tonne. Calculate the profit per tonne of each blend of tea. How muchof each blend should the company produce to maximize its profit (not gross but net returns) and what isthe maximum profit?2. A small manufacturing company produces three products, A, B and C at its plant. The parts of eachproduct must be manufactured, the products are then assembled and finally the finished products packagedfor distribution to wholesalers. Each unit of product A takes 5 hours to manufacture, 2 hours to assembleand 1 hour to package. The corresponding times for product B are 2 12hours, 1 hour and 40 minutes, andfor product C, 4 12hours, 1 12hours and 40 minutes. The manufacturer has up to 300 hours of manufacturingtime per week using skilled labour, 110 hours of assembling time, which uses semi-skilled labour, and 60hours of packaging time which uses unskilled labour. If the profits per unit A, B and C are $110, $60 and$90, respectively, how many units of each product should the manufacturer produce, assuming all productcan be sold?3. A farming company owns two farms, which differ in the growing of crops and their yields. Each farm has100 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 ata cost of $100 per acre. Farm B yields 350 bushels of wheat per acre at a cost of $85 per acre and 650bushels of corn per acre at a cost of $120 per acre. How should the crops be planted to minimise the cost?14. (MATH2070 only.) Formulate the dual problem of the primal in Question 1. Solve the dual problem ofQuestion 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,Xnj=1aijxj = bi, i = 1,..., m,can be found by minimising the sum of the absolute errors,Xmi=1bi?Xnj=1aijxj,with respect to xk, k = 1,..., n. This L1-approximation optimisation problem is equivalent to thefollowing LP problem with n + m decision variables xj, j = 1,..., n, and ei, i = 1,..., m.Minimise z =Xmj=1ejsubject to ei +Xnj=1aijxj? bi, i = 1,..., m,ei?Xnj=1aijxj??bi, i = 1,..., m.(a) Write down the dual problem using dual variables yi, i = 1,..., m, for the first m constraints andwi, i = 1,..., m, for the last m constraints. Show by eliminating the wi that the dual problem canbe simplified toMaximise v =Xmi=1biyisubject to Xmi=1aijyi =12Xmi=1aij, j = 1,..., n,0? yi? 1, i = 1,..., m.(b) Use the result of Part(a) to fit a straight line of the form y = ax + b to the six points data points inthe (x, y)-plane given by (?2,?2), (?1, 2), (1, 5), (2, 14), (3, 10), (5, 15) by minimising the sum of theabsolute errors at the six points (i.e. use L1 curve-fitting). Formulate the simplified dual LP problemof Part (a) for this curve-fitting problem and solve it using the MATLAB command linprog. (Thedecision variables m and b of the primal problem are given by -lambda.eqlin ? use the commandin 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 websitehttp://www.mathworks.com.au/help/optim/ug/linprog.html2(c) Using the same data points from part (b), fit another line that minimises the sum of squared errors,Xmi=1bi?Xnj=1aijxj2.Use Matlab?s backslash command \ with help file herehttp://www.mathworks.com.au/help/matlab/ref/mldivide.htmlPlot the data points and both of the straight lines you obtained in the same figure. Which point isthe outlier? Which line is less effected by the outlier, and hence a better fit? Use a plot to showwhat happens to the lines if you add 25 to the y-component of the outlier? Which fitting method isrobust to the outlier?3

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