Courses   <-  Sean Forman   <-  You Are Here Homework #1

MAT 2461, Operations Research, Sean Forman

Due Jan. 31, 2003

1. (5 pt.) For the following matrices in reduced-row echelon form, list the basic and non-basic variables and find the solution.

   $\left( \begin{array}{r r r r r} 1 & 0 & 1 & 2 & 5 \\ 0 & 1 & 1 & 0 & -4\\ 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 \end{array}\right)$ $\left( \begin{array}{r r r r r} 1 & 0 & 0 & 0 & -4 \\ 0 & 1 & 0 & 0 & 5\\ 0 & 0 & 1 & 0 & 2\\ 0 & 0 & 0 & 1 & 0 \end{array}\right)$ $\left( \begin{array}{r r r r r r r} 1 & 2 & 0 & 9 & 0 &-2 &-5\\ 0 & 0 & 1 &-3 & 0 & 6 & 4\\ 0 & 0 & 0 & 0 & 1 & 1 &12\\ \end{array}\right)$

2. (5 pt.) Place the following matrix in RREF, using the Gauss-Jordan Method.

   $\left( \begin{array}{r r r r r} 1 & 1 & 2 & 1 & 2\\ 2 & 2 & 3 & 3 & 9\\ 2 & 1 & 2 & 2 & 7\\ 1 & 1 & 1 & 1 & 4 \end{array}\right)$

3. (5 pt.) Find the graphical solution for the following problem.

   $\begin{array}{rrrrrrrr} \mbox{max} & & 150x_1 & + & 180x_2 \\ \par 3x_1 & + &... ...& + &6x_2 &\leq & 132 \\ x_1 & & &\geq & 0 \\ & & x_2 &\geq & 0 \end{array}$

4. (5 pt.) 3.5.5, you may wish to practice first on 3.5.3, which is a bit simpler.

5. (5 pt.) 3.11.5, assume that for the short-term line of credit, it can be paid off or added to as the year goes on. Here are variables, you will likely want to use. I believe you will want a set of constraints for the amount of short-term debt owed, and another set for the cash-on-hand.
   

   $L_t$ - long-term debt borrowed on Jan. 1, 2000.
   $L_i$ - short-term debt owed at the start of month $i$.
   $c_i$ - cash on hand at the end of month $i$.
   $b_i$ - amount of short-term debt added at the start of month $i$.
   $p_i$ - amount of short-term debt paid off at the start of month $i$.
   $f_i$ - cash-flow for month $i$.