Homework 01 - IEOR E4007 G Iyengar Sept 10th 2008...

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IEOR E4007 G. Iyengar Sept. 10th 2008 Homework #1 Due : Wednesday, September 24th, 2008. 1. Simple linear program Consider the following linear program (please solve this question by hand and not use any computer solver, you may use solver for all the other questions): max 2 x 1 + x 2 , subject to 12 x 1 + 3 x 2 6 , - 3 x 1 + x 2 7 , x 2 10 , x 1 ,x 2 0 . (a) Draw a graph of the constraints and shade in the feasible region. Label the vertices of this region with their coordinates. (b) Using the graph obtained in (a), find the optimal solution and the maximum value of the objective function. (c) What is the slack in each of the constraints? (d) Find the shadow prices on each of the constraints. (e) Find the ranges associated with the two coefficients of the objective function. (f) Find the righthand-side ranges for the three constraints. 2. Portfolio selection problem A portfolio manager in charge of a bank portfolio has $10 million to invest. The securities available for purchase, as well as their quality ratings, maturities and yields are shown below. Bond name Bond type Quality Yrs to Mat Yield to Mat After tax yield Moody’s Bank’s A Municipal Aa 2 9 4.3% 4.3% B Agency Aa 2 15 5.4% 2.7% C Government Aaa 1 4 5.0% 2.5% D Government Aaa 1 3 4.4% 2.2% E Municipal Ba 5 2 4.5% 4.5% The bank puts the following policy limitations on the portfolio manager’s actions: 1
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Government and Agency bonds must total at least $4 million.
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