1. Postscript on assignment 8 One E.R. faster than two separate ones 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 method First, 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 Solutions DT 00 08 010 020 46 65 6.673.33 80 100 160

12. Enumeration method SolutionsSlack in constraints DTABC 00203016 081260 010 0-4 0200-30-24 46400 650-30 6.673.33002.67 80068 100-506 160-20-180

13. Enumeration method SolutionsSlack in constraintsFeasible? DTABC 00203016Yes 081260Yes 010 0-4No 0200-30-24No 46400Yes 650-30No 6.673.33002.67Yes 80068 100-506No 160-20-180No

14. Enumeration method SolutionsSlack in constraintsFeasible?Profit DTABC 00203016Yes0 081260Yes32 010 0-4No 0200-30-24No 46400Yes36 650-30No 6.673.33002.67Yes33.3 80068Yes24 100-506No 160-20-180No

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. ConstraintsProductsIntersectionsCalculations 321053 642108,960 66924133,056 128125,97042,997,760

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.