The problem of optimizing a function over a feasible set that is not a linear program is more generally called a Mathematical Program (MP). The first three problems are exercises in transforming MPs into LPs. Problem 1 Consider the following MP minimize |x| subject to -3 ≤ x ≤ 2 Express it as an LP, and provide a geometric interpretation of the transformation. Problem 2 Consider the following MP minimize ( max {2x-4, |x|}) subject to -3≤ x ≤ 6 i) Express it as an LP with 1 additional variable and provide a geometric interpretation for the transformation. ii) Can you express it as an LP with 2 additional variables? Problem 3 Consider the following MP minimize |x1|+|x2| subject to -6 ≤ x1 ≤ 8 x1≤x2 Express it as an LP. Problem 4 Introduction to Solvertable (the setup file for solvertable is on server) Recall from recitation 1 the CCC problem: The Colorado Cattle Company (CCC) can purchase three types of raw feed ingredients from a wholesale distributor. The company's cattle have certain nutritional requirements with respect to fat, protein, calcium and iron. Each cow requires at least 10 units of calcium, no more than 7.5 units of fat, at least 12 units of iron and at least 15 units of protein per day. The table below shows the amount of fat, protein, calcium and iron in each pound of the three feed ingredients. Grade 1 feed costs $.25, grade 2, $.10, and grade 3, $.08 per pound. The cattle can be fed a mixture of the three types of raw feed. CCC would like to feed its herd as cheaply as possible. Page 1 of 3 Feed Ingredients (Units Per Pound) Grade 1 Grade 2 Grade 3 Calcium .7 .8 0 Iron .9 .8 .8 Protein .8 1.5 .9 Fat .5 .6 .4 Last time we formulated the problem and solved it using Excel Solver. Now assume that the Calcium requirement is not known. Solve the problem varying the calcium requirement from 5 to 15 in increments of 1 unit – using Solver Table. Problem 5 Introduction to CyTools (add-in file and instructions are on server) Suppose we would like to find a linear combination of the vectors ⎡− 3⎤ ⎢ 4 ⎥ and ⎣ ⎦ ⎡1⎤ ⎢3⎥ so that their combination sums to ⎣ ⎦ ⎡10 ⎤ ⎢20⎥ ⎣ ⎦ Use CyTools – Pivot Operation to solve this problem Problem 6. Portfolio Selection (Taken from Bradley, Hax and Magnanti, pg 11-13) A portfolio manager in charge of a bank portfolio has $10 million to invest. The securities available for purchase, as well as their respective quality ratings, maturities, and yields, are shown below: Bond Name A B C D E Quality Scales Moody’s Bank’s Municipal Aa 2 Agency Aa 2 Government Aaa 1 Government Aaa 1 Municipal Ba 5 Bond Type Years to Maturity 9 15 4 3 2 Yield to maturity 4.3% 5.4 5.0 4.4 4.5 After-tax yield 4.3% 2.7 2.5 2.2 4.5 The bank places the following policy limitations on the portfolio manager’s actions: 1. Government and agency bonds must total at least $4 million. 2. The average quality of the portfolio cannot exceed 1.4 on the bank’s quality scale. (Note that a low number on this scale means a high-quality bond.) Page 2 of 3 3. The average years to maturity of the portfolio must not exceed 5 years. Assuming that the objective of the portfolio manager is to maximize after-tax earnings, what bonds should he purchase? Formulate as a Linear Program • • Solve using solver • Use Solver table to vary allowable quality from 1 to 3 in increments of .2 and plot the results If it became possible to borrow up to $1 million at 5.5%, how should the selection be changed? Page 3 of 3