Linear Programming Intro 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 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. – The axes are constraints – You are not allowed to make a negative amount of any product.

Graph method: Using the profit function Pick an arbitrary profit number and set the profit equal to it. E.g. 3d + 4t = 12 Plot this on the graph Move this parallel to itself up or down until the line just touches a corner of the feasible area.

Graph method drawbacks How good a draftsman are you? Can’t work in three 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 good and bad You can do problems with more than two dimensions. The math grows rapidly as the number of activities and constraints grows. ConstraintsProductsIntersectionsCalculations , , ,97042,997,760

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.

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

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.