1. Chapter 24 – Multicriteria Capital Budgeting and Linear Programming u Linear programming is a mathematical procedure, usually carried out by computer software, to find an optimal combination to maximize a goal function subject to constraints.

2. Simple LP Problem We cannot invest more than $3,000 in division 1 or more than $2,000 in division 2. The amount invested in division 1 must be at least twice as great as the amount invested in division 2. x 1 and x 2 are the amounts invested in divisions 1 and 2. The LP model is Maximize.2x 1 +.3x 2 Subject tox 1 ≤ $3,000 x 2 ≤ $2,000 -x 1 + 2x 2 ≤ $0 x 1 ≥ $0 x 2 ≥ $0

3. Dual values u Each constraint has a dual value, which is typically computed by the same software that finds the original solution u Dual values tells us how much the objective function could be increased if we exceeded a particular constraint by one unit

4. Sensitivity Analysis u We often solve a linear programming problem several times, with constraints changed. As a result of this analysis, we might go back and look at ways to overcome a particular constraint

5. Solving LP problems with Excel u Excel Solver can be used to solve LP problems u You may need to install Solver because it is not automatically installed

6. Solver Illustration X1X1 X2X2 SumLimit Maximize0.20.30 Subject to 100<=3,000 010<=2,000 20<= 0 100=> 0 010 0 Variable values 00

7. Multiple Goals and Constraints u Managers can list multiple goals, such as NPV, sales growth, EPS growth, etc. u To put the goals in the objective function, each must be given a weight u Considerations can also be converted to constraints. We might have a constraint that EPS must increase at least 5% a year, for example

8. Capital rationing u Set the objective function as maximizing wealth as of some specified future date, given a fixed amount of capital available, and specified infusions of capital each year, if any. u Must specify reinvestment opportunity rates in future periods, at least in terms of estimated external opportunity rates

9. Other Programming Techniques u Integer programming u Goal programming u Chance-constrained programming u Quadratic programming