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Homework 02 - IEOR E4007 G Iyengar Homework#2 1 Simple...

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IEOR E4007 G. Iyengar Homework #2 1. Simple problems on simplex Consider a linear program with n = 4 variables and m = 2 constraints. Suppose the current basis B = { 1 , 2 } and the constraints are given by x 1 - 3 x 3 +3 x 4 = 6 x 2 - 8 x 3 +4 x 4 = 4 For each of the following set of objective vectors decide whether the current basis is optimal, if not then move to a better basis or declare the problem is unbounded. (a) max c = 1 1 - 3 - 1 0 (b) max c = 0 0 - 3 1 0 (c) min c = 1 1 - 3 8 0 2. Tinyco Cash Flow Management The cash position of the company is described as follows. Initial holding $200,000 in short-term bonds Demand for cash in months i = 1 , 2 , . . . , 5: - 100, 300, - 500, - 10, 30 (in thou- sands) Cash earns 0.5% per month and short-term bond earns 0.9% per month Tinyco can borrow up to $100,000 at 1% interest per month Transaction cost of 0.2% is charged for cash bonds. The bond coupon (interest) is paid in cash (no transaction costs) The goal of the company is to maximize cash position at the end of five months while meeting its liabilities. Formulate and solve the LP that solves this problem. 3. Exercise 3.12 & 3.18 from Optimization Methods in Finance You might want to read Section 3.2 and 3.3.2 before attempting to do this problem. (a) Exercise 3.12 (b) Exercise 3.18 1
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4. Formulating dual problems Formulate the duals of the following linear programs and also formulate the corre- sponding complementary slackness conditions. (a) max 44 x 1 - 3 x 2 + 15 x 3 + 56 x 4 s.t. x 1 + x 2 + x 3 + x 4 = 20 , x 1 - x 2 0 , 9 x 1 - 3 x 2 + x 3 - x 4 25 , x 1 , . . . , x 4 0 .
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