1 / 14

An Introduction to Linear Programming

An Introduction to Linear Programming. Chris L. Riley University of Mississippi February 15, 2007. What is Linear Programming?. Linear programming (LP) refers to a technique to solve a specific class of mathematical optimization problems.

maj
Télécharger la présentation

An Introduction to Linear Programming

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. An Introduction toLinear Programming Chris L. RileyUniversity of MississippiFebruary 15, 2007

  2. What is Linear Programming? • Linear programming (LP) refers to a technique to solve a specific class of mathematical optimization problems. • LP optimization problems have linear objective functions and constraints.

  3. The Relevance ofOptimization Problems to DSS • All of the following decisions can be considered optimization problems: • What crops an agribusiness should plant • How to efficiently assign employees to tasks • Deciding which publications to advertise in

  4. Characteristics of LP Problems • Some resources are only available in limited quantities • These resources are necessary in order to create deliverables: products, services, information • Resources can be used in multiple ways • Deliverables are associated with value in some way • Certain restrictions, or constraints, are imposed on resources, deliverables, values, etc. Ref: Decision Support Systems and Intelligent Systems, 7th Ed

  5. Assumptions for LP Problems • Value generated, or the return, from a particular allocation of resources can be measured in a common unit • All possible allocations are independent of each other • The return for an allocation is the sum of the returns for each activity • All inputs are known with certainty(with probability = 1) • Resources are used optimally Ref: Decision Support Systems and Intelligent Systems, 7th Ed

  6. Components of the Mathematical Model • A result variable (total return)The value of the result variable is the linear function that is to be optimized (maximized or minimized) • Decision variablesMost often the amount of resources to be used • Constraints or uncontrollable variables • Labor, budget, existing obligations, etc. Ref: Wikipedia

  7. Standard Form • Standard form is often used in describing a linear programming problem and consists of the following three parts: • A linear function to be maximized • e.g. maximize z=c1x1+c2x2 • Problem constraints of the following form • e.g. • a11x1 + a12x2 ≤ b1 • a21x1 + a22x2 ≤ b2 • Non-negative variables • e.g. • x1 ≥ 0 • x2 ≥ 0 Ref: Wikipedia

  8. An Example • Suppose that a farmer has a piece of farm land, say A square kilometers large, to be planted with either wheat or barley or some combination of the two. • The farmer has a limited permissible amount F of fertilizer and P of insecticide which can be used, each of which is required in different amounts per unit area for wheat (F1, P1) and barley (F2, P2). • Let S1 be the selling price of wheat, and S2 the price of barley. • If we denote the area planted with wheat and barley with x1 and x2 respectively, then the optimal number of square kilometers to plant with wheat vs barley can be expressed as a linear programming problem. Ref: Wikipedia

  9. Example in Standard Form • maximize • S1x1 + S2x2 (maximize the revenue) • subject to • x1 + x2 ≤ A (limit on total area) • F1x1 + F2x2 ≤ F (limit on fertilizer) • P1x1 + P2x2 ≤ P (limit on insecticide) • x1 ≥ 0, x2 ≥ 0 (cannot plant a negative area) Ref: Wikipedia

  10. Augmented Form • Most LP algorithms cannot solve a problem in standard form. • An augmented form that uses slack variables to remove the inequalities must be used. • Sophisticated LP packages take care of this automatically.

  11. Example in Augmented Form • maximize • S1x1 + S2x2 (maximize the revenue) • subject to • x1 + x2 + x3 = A (limit on total area) • F1x1 + F2x2 + x4 = F (limit on fertilizer) • P1x1 + P2x2 + x5 = P (limit on insecticide) • x1, x2, x3, x4, x5 ≥ 0 Ref: Wikipedia

  12. Solving LP Problems • Several algorithms and methods have been developed to solve LP problems • Simplex • Interior point methods • Projective method • Mehrotra predictor-corrector method

  13. Integer Programming • Integer Programming (IP)Decision variables are required to be integers • 0-1 IPDecision variables must be either 1 or 0 • Mixed Integer Programming (MIP)Some decision variables are required to be integers

  14. Sources of LP Solvers • Microsoft Excel • Mathematica • GNU Linear Programming Kit • Matlab (requires additional toolbox)

More Related