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

Linear Programming I HSPM J716

Linear Programming Optimization under constraint Linear constraints and objective function

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

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.

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 hours of assembly time are available. Constraint A: 2.5d + 1t <= 20 A desk takes 3 hours to buff; a table takes hours of buffing time are available. Constraint B: 3d + 3t <= 30 A desk takes 1 hour to crate; a table takes hours of crating time are available. Constraint C: 1d + 2t <= 16

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.

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.

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

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

Enumeration method Solutions DT

Enumeration method SolutionsSlack in constraints DTABC

Enumeration method SolutionsSlack in constraintsFeasible? DTABC Yes Yes No No 46400Yes No Yes No No

Enumeration method SolutionsSlack in constraintsFeasible?Profit DTABC Yes Yes No No 46400Yes No Yes Yes No No

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 , , ,97042,997,760

George B. Dantzig ( ) “The Father of Linear Programming”

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.

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.