Linear Programming

Student project for MA265



 

What is linear programing?


Linear programing is a mathematical method you can apply to find best possible results such as maximum profit or minimum cost. Linear programing practically use as one of the optimization problem solvers. In general, to solve a optimization problem, you need to have an objective (what do you want to solve? Do you want to maximize or minimize your function?) and constraints related to your function. Putting objective and constraints function together on a graph will generate a feasible region.

For example,

Objective function max -1/2x + 7

Constraints y=3x y=x-2

Linprog03.gif

(source: http://www.purplemath.com/modules/linprog.htm)

The shade area is the feasible region


There are many software that help you solve optimization problems. In this page, I will focus on how to use GAMS.


GAMS


GAMS can be accessed through ITAP software remote (https://goremote.itap.purdue.edu/Citrix/XenApp/auth/login.aspx)

GAMS is a software that we can use to solve optimization problems. It is simple yet powerful software. The following is a sample of how the code will look like on GAMS. The code before is trying to minimize the cost of a recycling company. There are methods to recycling and company wants to find the optimized way to operate.

We need to define our variables in this section

 variables
 objval    "objective function value"
 x1        "new wood pulp"
 x2        "recycled office paper"
 x3        "recycled newsprint"
 y1        "# of products produced by method 1"
 y2        "# of products produced by method 2"
 y3        "# of products produced by method 3"
 y4        "# of products produced by method 4"

We define the type of variables here

free variables

 objval;

positive variables

 x1,x2,x3,y1,y2,y3,y4;

equations

 obj         "total cost"
 method_1    "produce from method 1"
 method_2    "produce from method 2"
 method_3   "produce from method 3"
 rest_80    "process 1 has to bw lesser than or equal to 80"
 atleast_100 "the # of products must be at least 100";


This is where we code our objective function and constraints

obj ..

 objval =e= 100*x1+50*x2+20*x3;

method_1 ..

 x1 =e= 3*y1+y2+y3;

method_2 ..

 x2 =e= 4*y2+8*y4;

method_3 ..

 x3 =e= 12*y3;

rest_80 ..

 x1 =l= 80;

atleast_100 ..

 y1+y2+y3+y4 =g= 100;

Since our goal is to minimize the cost, we use "lp minimizing" function to minimize our 'objval'

model absmall /all/;

solve absmall using lp minimizing objval;


My experience with linear programing


I have learned linear programing from a class in school of industrial engineering, IE335. When the data is very big, for instance, calculations involving in traveling in the outer space, we need linear programing to help us break down huge problems to smaller problems. Linear programing and non-linear programming is still a very active area of research. We always want to find a quickest possible ways to optimize our operations. I found that linear programming is very useful and interesting.


Courses for linear programing


If you are interested more in Linear programing and would like to take a course. I have listed the courses that relates to Linear programming.

IE 335 - Operation Research - Optimization Introduction to deterministic optimization modeling and algorithms in operations research. Emphasis on formulation and solution of linear programs, networks flows, and integer programs. Typically offered Fall Spring.

CS 51500 - Numerical Linear Algebra. (was CS 515: Numerical Analysis of Linear Systems till 2009) Computational aspects of linear algebra; linear equations and matrices, direct and iterative methods; eigenvalues and eigenvectors of matrices; error analysis.

ECE 580 - Optimization Methods for Systems and Control.
Introduction to various methods of obtaining the extremum of a nondynamic or a dynamic system and their use in control system design. Linear programming, various search methods, nonlinear programming and dynamic programming. Various real-life applications are discussed, and appropriate case studies are investigated.

IE 535 - Linear Programming. 
Optimization of linear objective functions subject to linear constraints. Development of theory and algorithmic strategies for solving linear programming problems.



Further studies


If you are interested in reading about Linear programming this link will give you the list of textbooks you can look at.

List of textbooks


Stapel, Elizabeth. "Linear Programming: Introduction." Purplemath. Available from

   http://www.purplemath.com/modules/linprog.htm. Accessed 11 December 2011

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