1. Goal Programming Juan Vera 9/28/2009 1

2. 9/28/2009 2 A variation of Giapetto's wooden:Goal Programming Assume that Giapetto wants to produce at least 16 soldiers and 10 trains. Can this goal be satisfied?

3. A variation of Giapetto's wooden:Goal Programming x ≥ 16 y ≥ 10 3x + 2y ≤ 80 3x + y ≤50 x + 2y≤60 x , y ≥0 No point satisfies the resource constraints and meets the goals. What can be done in this situation? Consider x ≥ 16 and y ≥ 10 as goals from wich you can deviate 9/28/2009 3

4. Goal Programming In some situations, a decision maker may face multiple objectives, and there may be no point in an LP’s feasible region satisfying all objectives. In such a case, how can the decision maker choose a satisfactory decision? Goal programming is one technique that can be used.

5. Deviational variables si+ = amount by which we exceed the ith goal si- = amount by which we are under the ith goal A variation of Giapetto's wooden:Goal Programming x = 16 x - s+ + s- = 16 s+, s- ≥ 0

6. A variation of Giapetto's wooden:Goal Programming It is impossible to meet both goals. Identify, for each goal, a cost for failing to meet the goal. Formulate an LP minimizing cost of deviating from goals. Goal Programming formulation Original problem: find x and y such that x ≥ 16 y ≥ 10 3x + 2y ≤ 80 3x + y ≤50 x + 2y≤60 x , y ≥0 WORK IT OUT!!!!

7. A variation of Giapetto's wooden:Goal Programming If cost are non-negative solutions should be in this region

8. Pre-emptive goal programming Instead of assigning price to goals assign priority. Minimize deviation according to priority. For example assume x has more priority than y. First, minimize deviation from x-goal: Solution (16, 0, 0, 0, 0, 10)

9. Pre-emptive goal programming- cont. Fix x = … and minimize deviation from y-goal. Solution (16, 2, 0, 0, 0, 8) Exercise: solve assuming y has more priority than x

10. Can also try to minimize the maximum deviation min max(s1- ,s2- ) st x -s1+ + s1- = 16 y -s2+ +s2- = 10 3x + 2y ≤ 80 3x + y ≤50 x + 2y≤60 x , y , s1+ , s1- , s2+ , s2-≥0 Solution (16, 2, 0, 0, 0, 8) Write as WORK IT OUT!!!!

11. Can also try other measures of deviation min f(s1- ,s1+,s2- s2+) st x -s1+ + s1- = 16 y -s2+ +s2- = 10 3x + 2y ≤ 80 3x + y ≤50 x + 2y≤60 x , y , s1+ , s1- , s2+ , s2-≥0 For example: f(s1- ,s1+,s2- s2+) = (s1- )2 + (s1+)2 + (s2- )2 +(s2+)2 or just, f(s1- ,s1+,s2- s2+) = (s1- )2 + (s2- )2 are used in practice (no an LP, but still not that difficult to solve!). In contrast if you allow any quadratic, it may become a very hard problem

12. An application:Radiation Delivery Improvements of delivery of radiation New field: tomotherapy IMRT

13. Linear Programming Model for radiation therapy This two parameters are set by physicians (minimum and maximum levels of radiation on tumor and critical tissue) Usually this LP is not feasible What to do? LT UC Xij Xij Xij Xij Goal Programming: Treat those constrains as goals

14. Solution obtained NLP solution (sum of squares) LP solution (weighted combination of deviations) 9/28/2009 14