Postscript on assignment 8

1. Postscript on assignment 8 • One E.R. faster than two separate ones Emergency Dept. Emergency Dept. Emergency Dept.

2. Linear Programming I HSPM J716

3. Linear Programming • Optimization under constraint • Linear constraints and objective function

4. Elements of a Linear Programming Manufacturing Problem • Things you can make or do in different amounts. • Constraints • Tell you how much you get from different combinations of resources • Tell you how much you have of each resource. • Objective function • Assigns a value to what you make • Your objective is to maximize this value

5. What Linear Implies • No increasing or diminishing returns in the use of the resources. • Everything just multiplies and adds. • The profit or revenue is linear, too. How much you make is price times quantity. No declining demand curve.

6. Translate the words into math • Profit is $3 per desk and $4 per table. • Objective function Profit = 3d + 4t • A desk takes 2½ hours to assemble; a table takes 1. • 20 hours of assembly time are available. • Constraint A: 2.5d + 1t <= 20 • A desk takes 3 hours to buff; a table takes 3. • 30 hours of buffing time are available. • Constraint B: 3d + 3t <= 30 • A desk takes 1 hour to crate; a table takes 2. • 16 hours of crating time are available. • Constraint C: 1d + 2t <= 16

7. Graph methodFirst, find the feasible area. • Each product is assigned to an axis. • Plot the constraints as equalities. • Draw a line for each constraint. • The feasible area is the polygon formed by the axes and the lowest constraints*. • * in a maximization problem • The axes are constraints, too. You cannot make a negative amount of any product.

8. Once you have the feasible area,use the profit function • Pick an arbitrary profit number and set the profit equal to it. • E.g. 3d + 4t = 12 • Plot this line on the graph • Move this line parallel to itself up or down until it just touches a corner of the feasible area. • That corner is your optimum.

9. Graph method drawbacks • How good a draftsman are you? • Hard to do in three dimensions. Impossible in four or more dimensions.

10. Enumeration method • Find all the intersections • Of the constraints • And the axes • Test each for feasibility • Choose the feasible intersection with the highest profit.

11. Enumeration method

12. Enumeration method

13. Enumeration method

14. Enumeration method

15. Enumeration method good and bad • You can do problems with more than two dimensions. • You get exact answers. • The math grows rapidly with the number of activities and constraints.

16. George B. Dantzig (1914-2005)“The Father of Linear Programming”

17. Simplex Method • A closed shape with flat sides is a “simplex.” • The simplex method starts with a corner of the feasible area that is easy to find. • Then it crawls along an edge to another corner. It picks the direction that makes profit go up the fastest. • It keeps going until it finds a corner where any move lowers profit. • Shortcuts the enumeration method. A local maximum is a global maximum.

18. Shadow price • How much more money you could make if you had one more unit of a resource • That’s the shadow price for that resource • If you could buy one more unit of a resource, the most you’d be willing to pay would be the shadow price.