WOOD 492 MODELLING FOR DECISION SUPPORT Lecture 6 LP Assumptions.

Slides:



Advertisements
Similar presentations
Agenda of Week III. LP I LP Standardization Optimization LP intro Week Definition Basic assumptions Example General form Standard form Objective.
Advertisements

WOOD 492 MODELLING FOR DECISION SUPPORT Lecture 17 Integer Programming.
IENG313 Operation Research I
WOOD 492 MODELLING FOR DECISION SUPPORT Lecture 12 Duality Theory.
WOOD 492 MODELLING FOR DECISION SUPPORT Lecture 15 Forest Planning.
Optimization problems using excel solver
1 Material to Cover  relationship between different types of models  incorrect to round real to integer variables  logical relationship: site selection.
Introduction to Mathematical Programming
Introduction to Mathematical Programming Matthew J. Liberatore John F. Connelly Chair in Management Professor, Decision and Information Technologies.
Linear programming: lp_solve, max flow, dual CSC 282 Fall 2013.
Linear Programming Problem. Introduction Linear Programming was developed by George B Dantzing in 1947 for solving military logistic operations.
Lesson 08 Linear Programming
Linear Programming. Introduction: Linear Programming deals with the optimization (max. or min.) of a function of variables, known as ‘objective function’,
Linear Programming.
Planning with Linear Programming
Linear Programming Models & Case Studies
WOOD 492 MODELLING FOR DECISION SUPPORT Lecture 10 Introduction to Sensitivity Analysis.
Chapter 2: Modeling with Linear Programming & sensitivity analysis
Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India LINEAR PROGRAMMING GRAPHIC METHOD 2 CHAPTER.
Basic Linear Programming Concepts Lecture 2 (4/1/2015)
Operations Management
Operations Management
Basic LP Problem McCarl and Spreen Chapter 2 LP problem is linear form of Mathematical Program This formulation may also be expressed in matrix notation.
Linear and Integer Programming Models
Linear Goal Programming
Linear Programming General Form of an LP Model. Linear Programming General Form of an LP Model where the c’s, a’s and b’s are constants determined from.
INTRODUCTION TO LINEAR PROGRAMMING
Operations Management - 5 th Edition Chapter 13 Supplement Roberta Russell & Bernard W. Taylor, III Linear Programming.
1 Ardavan Asef-Vaziri Nov-2010Theory of Constraints 1- Basics Purchased Part $5 / unit RM1 $20 per unit RM2 $20 per unit RM3 $25 per unit $90 / unit 110.
9/1 More Linear Programming Collect homework Roll call Review homework Lecture - More LP Small Groups Lecture - Start using MS Excel Assign Homework.
WOOD 492 MODELLING FOR DECISION SUPPORT Lecture 2 Introduction to Linear Programming.
1-1 Introduction to Optimization and Linear Programming Chapter 1.
Introduction to Linear Programming
Chapter 3 Introduction to Optimization Modeling
WOOD 492 MODELLING FOR DECISION SUPPORT Lecture 1 Introduction to Operations Research.
Product Mix Problem Monet company makes four types of frames.
Linear Programming Models Tran Van Hoai Faculty of Computer Science & Engineering HCMC University of Technology Tran Van Hoai.
Linear Programming Topics General optimization model LP model and assumptions Manufacturing example Characteristics of solutions Sensitivity analysis Excel.
1 DSCI 3023 Linear Programming Developed by Dantzig in the late 1940’s A mathematical method of allocating scarce resources to achieve a single objective.
BUSINESS MATHEMATICS & STATISTICS. LECTURE 45 Planning Production Levels: Linear Programming.
WOOD 492 MODELLING FOR DECISION SUPPORT Lecture 4-5 LP Formulation Example and Excel Solver.
WOOD 492 MODELLING FOR DECISION SUPPORT Lecture 11 Sensitivity Analysis.
WOOD 492 MODELLING FOR DECISION SUPPORT Lecture 9 Intro to Sensitivity Analysis.
1 The Dual in Linear Programming In LP the solution for the profit- maximizing combination of outputs automatically determines the input amounts that must.
WOOD 492 MODELLING FOR DECISION SUPPORT Lecture 3 Basics of the Simplex Algorithm.
WOOD 492 MODELLING FOR DECISION SUPPORT Lecture 7 LP Formulation Examples.
1 INTRODUCTION TO LINEAR PROGRAMMING CONTENTS Introduction to Linear Programming Applications of Linear Programming Reference: Chapter 1 in BJS book.
WOOD 492 MODELLING FOR DECISION SUPPORT Lecture 18 Branch and Bound Algorithm.
Arben Asllani University of Tennessee at Chattanooga Business Analytics with Management Science Models and Methods Chapter 2 Introduction to Linear Programming.
ENGM 631 Optimization Ch. 3: Introduction to Linear Programming.
CDAE Class 12 Oct. 4 Last class: 2. Review of economic and business concepts Today: 3. Linear programming and applications Quiz 3 (sections 2.5 and.
LINEAR PROGRAMMING.
OPSM 301 Operations Management Class 11: Linear Programming using Excel Koç University Zeynep Aksin
MCCARL AND SPREEN TEXT CH. 2 T Y/MCCARL-BRUCE/BOOKS.HTM Lecture 2: Basic LP Formulation.
WOOD 492 MODELLING FOR DECISION SUPPORT Lecture 14 Sensitivity Analysis.
Class 10: Introduction to Linear Programming
Linear Programming Topics General optimization model
Problem 1 Demand Total There are 20 full time employees, each can produce 10.
Linear Programming – Introduction
McCarl and Spreen Chapter 2
Linear Programming Topics General optimization model
Lecture 5 – Integration of Network Flow Programming Models
Linear Programming Topics General optimization model
Linear Programming Topics General optimization model
The application of mathematics and the scientific
Basic Linear Programming Concepts
Spreadsheet Modeling & Decision Analysis
Linear Programming Introduction.
Operations Research Models
Linear Programming Introduction.
Presentation transcript:

WOOD 492 MODELLING FOR DECISION SUPPORT Lecture 6 LP Assumptions

Last Week Solving LPs with the Excel Solver LP Matrix format Sept 17, 2012Wood Saba Vahid2

Assumptions of LP For a system to be modelled with an LP, 4 assumptions must hold: Proportionality, Additivity, Divisibility, and Certainty –Proportionality: Contribution of each activity (decision variable) to the Obj. Fn. is proportional to its value (represented by its coefficient in the Obj. Fn.), e.g. Z=3x 1 +2x 2, when x 1 is increased, its contribution to the Obj. is always increased three-fold (3x 1 ). –invalid assumption: e.g. manufacturing a product has a startup-cost: If there is no products made (x=0), the total profits would be zero (Z=c.x=0), but if any products are made (x>0), the profits are not proportional to the volume of products (Z=c.x-d), where d is the start-up cost. Sept 17, 2012Wood Saba Vahid3

Assumptions of LP- Cont’d –Additivity: Every function in an LP (Obj. Fn. or the constraints) is the linear sum of individual contributions of the respective activities (decision variables) e.g. x 1 +12x 2 <=100, is the sum of two linear functions, each showing the level of contribution of a variable (x1 or x2) to the constraint –invalid assumption: e.g. the products are complementary profits of the combined production is more than the sum of the individual production profits (Z=cx 1 +dx 2 + x 1.x 2 ) Sept 17, 2012Wood Saba Vahid4 Extra, nonlinear term

Assumptions of LP- Cont’d –Divisibility: Activities can be run at fractional level, i.e., decision variables can have any level (not just integer values). e.g. x1=33.3, x2=0.01 –invalid assumption: no fractional values for decision variables allowed e.g. assigning workers to different processes, scheduling shifts, building roads. –Certainty: Parameter values (coefficients in the functions) are known with certainty e.g. required hours to produce each product is known with certainty. –invalid assumption: e.g. when production costs are not known with certainty This happens commonly and therefore sensitivity analysis is an important part of any LP solution analysis. Sept 17, 2012Wood Saba Vahid5

Examples of Objective functions Max profit Min costs Max utility Max turnover Max Return on Investment Max Net Present Value Min number of employees Min redundancy Max customer satisfaction Sept 17, 2012Wood Saba Vahid6

Examples of LP constraints Upper & lower bounds (on raw material or products) Productive capacity Raw material availability Marketing demands & limitations Material balance (for balancing the input-output conversions within the model) Production ratio (link between the production of two or more products) Sept 17, 2012Wood Saba Vahid7

Example: Cut-Fill areas for road building In order to even out the road: earth should be transferred from cut areas (C1-C3) or borrow pit to Fill areas (F1-F4) or waste pit. Sept 17, 2012Wood Saba Vahid8

Example: Cut-Fill areas What is our objective? –Minimize total earth transfer costs ($) What are our decision variables? –How much earth (m3) to transfer from each cut area or borrow pit to each fill area or waste site What are our constraints? –The available volume of earth in the cut areas (m3) –The required volume of earth for fill areas (m3) Naming our variables: –C1F1: volume of earth (m3) transferred from C1 to F1 –C1W: volume of earth (m3) transferred from C1 to waste site –BF1: volume of earth (m3) transferred from the borrow pit to F1 Sept 17, 2012Wood Saba Vahid9 Cut-Fill Example

Next Class More formulation examples Preview of the lab Sept 17, 2012Wood Saba Vahid10