Linear Programming: Basic Concepts
|
|
- Brittney Chapman
- 7 years ago
- Views:
Transcription
1 Linear Programming: Basic Concepts
2 Table of Contents Three Classic Applications of LP The Wyndor Glass Company Product Mix Problem Formulating the Wyndor Problem on a Spreadsheet The Algebraic Model for Wyndor The Graphical Method Applied to the Wyndor Problem Using the Excel Solver with the Wyndor Problem A Minimization Example The Profit & Gambit Co. Introduction to Linear Programming The Graphical Method and Properties of LP Solutions 2
3 Three Classic Applications of LP Product Mix at Ponderosa Industrial Considered limited resources, and determined optimal mix of plywood products. Increased overall profitability of company by 20%. Personnel Scheduling at United Airlines Designed work schedules for all employees at a location to meet service requirements most efficiently. Saved $ million annually. Planning Supply, Distribution, and Marketing at Citgo Petroleum Corporation The SDM system uses LP to coordinate the supply, distribution, and marketing of each of Citgo s major products throughout the United States. The resulting reduction in inventory added $1 million annually to Citgo s profits. 3
4 Wyndor Glass Co. Product Mix Problem Wyndor has developed the following new products: An -foot glass door with aluminum framing. A -foot by -foot double-hung, wood-framed window. The company has three plants Plant 1 produces aluminum frames and hardware. Plant 2 produces wood frames. Plant 3 produces glass and assembles the windows and doors. Questions: 1. Should they go ahead with launching these two new products? 2. If so, what should be the product mix?
5 Algebraic Model for Wyndor Glass Co. Let D = the number of doors to produce W = the number of windows to produce Maximize P = $300D + $500W subject to D 2W 12 3D + 2W 1 and D 0, W 0. 5
6 Graphing the Product Mix W Production rate (units per week) for windows A product mix of D = and W = (, ) A product mix of D = 2 and W = 3 (2, 3) 1 Origin Production rate (units per week) for doors -1 D -2
7 Graph Showing Constraints: D 0 and W 0 W 2 Production rate for windows 0 2 Production rate for doors D 7
8 Nonnegative Solutions Permitted by D W D = 2 Production rate for windows 0 2 D Production rate for doors
9 Nonnegative Solutions Permitted by 2W 12 Production rate for windows W 2 W = Production rate for doors D 9
10 Boundary Line for Constraint 3D + 2W 1 Production rate for windows W (0, 9) (1, 7 1_ ) 2 (2, ) 3 D + 2 W = 1 (3, 1_ ) 2 (, 3) 2 (5, 1 1_ ) 2 (, 0) 0 2 Production rate for doors D
11 Changing Right-Hand Side Creates Parallel Constraint Boundary Lines Production rate for windows W 12 3D + 2W = 2 3D + 2W = 1 2 3D + 2W = Production rate for doors D 11
12 Nonnegative Solutions Permitted by 3D + 2W 1 Production rate for windows W 3D + 2W = Production rate for doors D 12
13 Graph of Feasible Region Production rate for windows W 3 D + 2 W = 1 D = 2 W =12 Feasible 2 region 0 2 Production rate for doors D 13
14 Objective Function (P = 1,500) Production rate for windows W P = 1500 = 300D + 500W Feasible region Production rate for doors D 1
15 Finding the Optimal Solution Production rate W for windows P = 300 = 300D + 500W P = 3000 = 300D + 500W Optimal solution (2, ) P = 1500 = 300D + 500W Feasible region Production rate for doors D 15
16 Summary of the Graphical Method Draw the constraint boundary line for each constraint. Use the origin (or any point not on the line) to determine which side of the line is permitted by the constraint. Find the feasible region by determining where all constraints are satisfied simultaneously. Determine the slope of one objective function line. All other objective function lines will have the same slope. Move a straight edge with this slope through the feasible region in the direction of improving values of the objective function. Stop at the last instant that the straight edge still passes through a point in the feasible region. This line given by the straight edge is the optimal objective function line. A feasible point on the optimal objective function line is an optimal solution. 1
17 Developing a Spreadsheet Model Step #1: Data Cells Enter all of the data for the problem on the spreadsheet. Make consistent use of rows and columns. It is a good idea to color code these data cells (e.g., light blue) B C D E F G Tables Chairs Profit $20.00 $15.00 Bill of Materials Available Large Bricks 2 1 Small Bricks
18 Developing a Spreadsheet Model Step #2: Changing Cells Add a cell in the spreadsheet for every decision that needs to be made. If you don t have any particular initial values, just enter 0 in each. It is a good idea to color code these changing cells (e.g., yellow with border) B C D E F G Doors Windows Unit Profit $300 $500 Hours Available Hours Used Per Unit Produced Plant Plant Plant Doors Windows Units Produced 0 0 1
19 Developing a Spreadsheet Model Step #3: Target Cell Develop an equation that defines the objective of the model. Typically this equation involves the data cells and the changing cells in order to determine a quantity of interest (e.g., total profit or total cost). It is a good idea to color code this cell (e.g., orange with heavy border) B C D E F G Doors Windows Unit Profit $300 $500 Hours Available Hours Used Per Unit Produced Plant Plant Plant Doors Windows Total Profit Units Produced 1 1 $ G Total Profit =SUMPRODUCT(UnitProfit,UnitsProduced) 19
20 Developing a Spreadsheet Model Step #: Constraints For any resource that is restricted, calculate the amount of that resource used in a cell on the spreadsheet (an output cell). Define the constraint in three consecutive cells. For example, if Quantity A Quantity B, put these three items (Quantity A,, Quantity B) in consecutive cells B C D E F G Doors Windows Unit Profit $300 $500 Hours Hours Hours Used Per Unit Produced Used Available Plant <= 1 Plant <= 12 Plant <= 1 Doors Windows Total Profit Units Produced 1 1 $ E Hours Used =SUMPRODUCT(C7:D7,UnitsProduced) =SUMPRODUCT(C:D,UnitsProduced) =SUMPRODUCT(C9:D9,UnitsProduced) 20
21 A Trial Solution B C D E F G Doors Windows Unit Profit $300 $500 Hours Hours Hours Used Per Unit Produced Used Available Plant <= 1 Plant <= 12 Plant <= 1 Doors Windows Total Profit Units Produced 3 $2,700 The spreadsheet for the Wyndor problem with a trial solution ( doors and 3 windows) entered into the changing cells. 21
22 Identifying the Target Cell and Changing Cells Choose the Solver from the Tools menu. Select the cell you wish to optimize in the Set Target Cell window. Choose Max or Min depending on whether you want to maximize or minimize the target cell. Enter all the changing cells in the By Changing Cells window B C D E F G Doors Windows Unit Profit $300 $500 Hours Hours Hours Used Per Unit Produced Used Available Plant <= 1 Plant <= 12 Plant <= 1 Doors Windows Total Profit Units Produced 1 1 $00 22
23 Adding Constraints To begin entering constraints, click the Add button to the right of the constraints window. Fill in the entries in the resulting Add Constraint dialogue box B C D E F G Doors Windows Unit Profit $300 $500 Hours Hours Hours Used Per Unit Produced Used Available Plant <= 1 Plant <= 12 Plant <= 1 Doors Windows Total Profit Units Produced 1 1 $00 23
24 The Complete Solver Dialogue Box 2
25 Some Important Options Click on the Options button, and click in both the Assume Linear Model and the Assume Non-Negative box. Assume Linear Model tells the Solver that this is a linear programming model. Assume Non-Negative adds nonnegativity constraints to all the changing cells. 25
26 The Solver Results Dialogue Box 2
27 The Optimal Solution B C D E F G Doors Windows Unit Profit $300 $500 Hours Hours Hours Used Per Unit Produced Used Available Plant <= 1 Plant <= 12 Plant <= 1 Doors Windows Total Profit Units Produced 2 $3,00 27
28 The Profit & Gambit Co. Management has decided to undertake a major advertising campaign that will focus on the following three key products: A spray prewash stain remover. A liquid laundry detergent. A powder laundry detergent. The campaign will use both television and print media The general goal is to increase sales of these products. Management has set the following goals for the campaign: Sales of the stain remover should increase by at least 3%. Sales of the liquid detergent should increase by at least 1%. Sales of the powder detergent should increase by at least %. Question: how much should they advertise in each medium to meet the sales goals at a minimum total cost? 2
29 Algebraic Model for Profit & Gambit Let TV = the number of units of advertising on television PM = the number of units of advertising in the print media Minimize Cost = TV + 2PM (in millions of dollars) subject to Stain remover increased sales: PM 3 Liquid detergent increased sales: 3TV + 2PM 1 Powder detergent increased sales: TV + PM and TV 0, PM 0. 29
30 Applying the Graphical Method Amount of print media advertising PM Feasible region PM = 3 2 -TV + PM = 3 TV + 2 PM = Amount of TV advertising TV 30
31 The Optimal Solution PM Cost = 15 = TV + 2 PM Feasible region Cost = = TV + 2 PM (,3) optimal solution Amount of TV advertising TV 31
32 Summary of the Graphical Method Draw the constraint boundary line for each constraint. Use the origin (or any point not on the line) to determine which side of the line is permitted by the constraint. Find the feasible region by determining where all constraints are satisfied simultaneously. Determine the slope of one objective function line. All other objective function lines will have the same slope. Move a straight edge with this slope through the feasible region in the direction of improving values of the objective function. Stop at the last instant that the straight edge still passes through a point in the feasible region. This line given by the straight edge is the optimal objective function line. A feasible point on the optimal objective function line is an optimal solution. 32
33 Profit & Gambit Co. Spreadsheet Model B C D E F G Television Print Media Unit Cost ($millions) 1 2 Increased Minimum Increase in Sales per Unit of Advertising Sales Increase Stain Remover 0% 1% 3% >= 3% Liquid Detergent 3% 2% 1% >= 1% Powder Detergent -1% % % >= % Total Cost Television Print Media ($millions) Advertising Units 3 33
34 A Production Problem Weekly supply of raw materials: Products: Small Bricks Large Bricks Table Profit = $20 / Table Chair Profit = $15 / Chair 3
35 Linear Programming Linear programming uses a mathematical model to find the best allocation of scarce resources to various activities so as to maximize profit or minimize cost. Let T = Number of tables to produce C = Number of chairs to produce Maximize Profit = ($20)T + ($15)C subject to 2T + C large bricks 2T + 2C small bricks and T 0, C 0. 35
36 Graphical Representation Tables Chairs + 2 Tables = Small Bricks 2 1 Chairs + 2 Tables = Large Bricks Chairs 3
37 Components of a Linear Program Data Cells Changing Cells ( Decision Variables ) Target Cell ( Objective Function ) Constraints 37
38 Four Assumptions of Linear Programming Linearity Divisibility Certainty Nonnegativity 3
39 When is a Spreadsheet Model Linear? All equations (output cells) must be of the form = ax + by + cz + where a, b, c are constants (data cells) and x, y, z are changing cells. Suppose C1:C are changing cells and D1:D are data cells. Which of the following can be part of an LP? SUMPRODUCT(D1:D, C1:C) SUM(C1:C) C1 * SUM(C:C) SUMPRODUCT(C1:C3, C:C) IF(C1 > 3, 2*C3 + C, 3*C3 + C5) IF(D1 > 3, C1, C2) MIN(C1, C2) MIN(D1, D2) * C1 ROUND(C1) 39
40 Why Use Linear Programming? Linear programs are easy (efficient) to solve The best (optimal) solution is guaranteed to be found (if it exists) Useful sensitivity analysis information is generated Many problems are essentially linear 0
41 Developing a Spreadsheet Model Step #1: Data Cells Enter all of the data for the problem on the spreadsheet. Make consistent use of rows and columns. It is a good idea to color code these data cells (e.g., light blue) B C D E F G Tables Chairs Profit $20.00 $15.00 Bill of Materials Available Large Bricks 2 1 Small Bricks 2 2 1
42 Developing a Spreadsheet Model Step #2: Changing Cells Add a cell in the spreadsheet for every decision that needs to be made. If you don t have any particular initial values, just enter 0 in each. It is a good idea to color code these changing cells (e.g., yellow with border) B C D E F G Tables Chairs Profit $20.00 $15.00 Bill of Materials Available Large Bricks 2 1 Small Bricks 2 2 Tables Chairs Production Quantity: 0 0 2
43 Developing a Spreadsheet Model Step #3: Target Cell Develop an equation that defines the objective of the model. Typically this equation involves the data cells and the changing cells in order to determine a quantity of interest (e.g., total profit or total cost). It is a good idea to color code this cell (e.g., orange with heavy border) B C D E F G Tables Chairs Profit $20.00 $15.00 Bill of Materials Available Large Bricks 2 1 Small Bricks 2 2 Tables Chairs Total Profit Production Quantity: 1 0 $ Total Profit =SUMPRODUCT(C:D,C11:D11) G 3
44 Developing a Spreadsheet Model Step #: Constraints For any resource that is restricted, calculate the amount of that resource used in a cell on the spreadsheet (an output cell). Define the constraint in three consecutive cells. For example, if Quantity A Quantity B, put these three items (Quantity A,, Quantity B) in consecutive cells. Note the use of relative and absolute addressing to make it easy to copy formulas in column E B C D E F G Tables Chairs Profit $20.00 $15.00 Bill of Materials Total Used Available Large Bricks <= Small Bricks 2 2 <= Tables Chairs Total Profit Production Quantity: 1 1 $ E Total Used =SUMPRODUCT(C7:D7,$C$11:$D$11) =SUMPRODUCT(C:D,$C$11:$D$11)
45 Defining the Target Cell Choose the Solver from the Tools menu. Select the cell you wish to optimize in the Set Target Cell window. Choose Max or Min depending on whether you want to maximize or minimize the target cell B C D E F G Tables Chairs Profit $20.00 $15.00 Bill of Materials Total Used Available Large Bricks <= Small Bricks 2 2 <= Tables Chairs Total Profit Production Quantity: 1 1 $
46 Identifying the Changing Cells Enter all the changing cells in the By Changing Cells window. You may either drag the cursor across the cells or type the addresses. If there are multiple sets of changing cells, separate them by typing a comma B C D E F G Tables Chairs Profit $20.00 $15.00 Bill of Materials Total Used Available Large Bricks <= Small Bricks 2 2 <= Tables Chairs Total Profit Production Quantity: 1 1 $35.00
47 Adding Constraints To begin entering constraints, click the Add button to the right of the constraints window. Fill in the entries in the resulting Add Constraint dialogue box B C D E F G Tables Chairs Profit $20.00 $15.00 Bill of Materials Total Used Available Large Bricks <= Small Bricks 2 2 <= Tables Chairs Total Profit Production Quantity: 1 1 $
48 Some Important Options Click on the Options button, and click in both the Assume Linear Model and the Assume Non-Negative box. Assume Linear Model tells the Solver that this is a linear programming model. Assume Non-Negative adds nonnegativity constraints to all the changing cells.
49 The Solution After clicking Solve, you will receive one of four messages: Solver found a solution. All constraints and optimality conditions are satisfied. Set cell values did not converge. Solver could not find a feasible solution. Conditions for Assume Linear Model are not satisfied B C D E F G Tables Chairs Profit $20.00 $15.00 Bill of Materials Total Used Available Large Bricks 2 1 <= Small Bricks 2 2 <= Tables Chairs Total Profit Production Quantity: 2 2 $
50 The Graphical Method for Solving LP s Formulate the problem as a linear program Plot the constraints Identify the feasible region Draw an imaginary line parallel to the objective function (Z = a) Find the optimal solution 50
51 Example #1 Maximize Z = 3x 1 + 5x 2 subject to x 1 2x x 1 + 2x 2 1 and x 1 0, x 2 0. x x 1 51
52 Example #2 Minimize Z = 15x x 2 subject to x 1 +2x 2 2x 1 3x 2 x 1 + x 2 and x 1 0, x 2 0. x x 1 52
53 Example #3 Maximize Z = x 1 + x 2 subject to x 1 +2x 2 = x 1 x 2 0 and x 1 0, x 2 0. x x 1 53
54 Properties of Linear Programming Solutions An optimal solution must lie on the boundary of the feasible region. There are exactly four possible outcomes of linear programming: A unique optimal solution is found. An infinite number of optimal solutions exist. No feasible solutions exist. The objective function is unbounded (there is no optimal solution). If an LP model has one optimal solution, it must be at a corner point. If an LP model has many optimal solutions, at least two of these optimal solutions are at corner points. 5
55 Example # (Multiple Optimal Solutions) Minimize Z = x 1 + x 2 subject to x 1 2x x 1 + 2x 2 1 and x 1 0, x 2 0. x x 1 55
56 Example #5 (No Feasible Solution) Maximize Z = 3x 1 + 5x 2 subject to x 1 5 x 2 3x 1 + 2x 2 1 and x 1 0, x 2 0. x x 1 5
57 Example # (Unbounded Solution) Maximize Z = 5x x 2 subject to x 1 5 2x 1 x 2 2 and x 1 0, x 2 0. x x 1 57
58 The Simplex Method Algorithm 1. Start at a feasible corner point (often the origin). 2. Check if adjacent corner points improve the objective function: a) If so, move to adjacent corner and repeat step 2. b)if not, current corner point is optimal. Stop. x x 1 5
Linear Programming. Solving LP Models Using MS Excel, 18
SUPPLEMENT TO CHAPTER SIX Linear Programming SUPPLEMENT OUTLINE Introduction, 2 Linear Programming Models, 2 Model Formulation, 4 Graphical Linear Programming, 5 Outline of Graphical Procedure, 5 Plotting
More information3 Introduction to Linear Programming
3 Introduction to Linear Programming 24 The development of linear programming has been ranked among the most important scientific advances of the mid-20th century, and we must agree with this assessment.
More informationModule1. x 1000. y 800.
Module1 1 Welcome to the first module of the course. It is indeed an exciting event to share with you the subject that has lot to offer both from theoretical side and practical aspects. To begin with,
More informationSolving Linear Programs in Excel
Notes for AGEC 622 Bruce McCarl Regents Professor of Agricultural Economics Texas A&M University Thanks to Michael Lau for his efforts to prepare the earlier copies of this. 1 http://ageco.tamu.edu/faculty/mccarl/622class/
More informationUSING EXCEL 2010 TO SOLVE LINEAR PROGRAMMING PROBLEMS MTH 125 Chapter 4
ONE-TIME ONLY SET UP INSTRUCTIONS Begin by verifying that the computer you are using has the Solver Add-In enabled. Click on Data in the menu across the top of the window. On the far right side, you should
More informationChapter 5. Linear Inequalities and Linear Programming. Linear Programming in Two Dimensions: A Geometric Approach
Chapter 5 Linear Programming in Two Dimensions: A Geometric Approach Linear Inequalities and Linear Programming Section 3 Linear Programming gin Two Dimensions: A Geometric Approach In this section, we
More informationEdExcel Decision Mathematics 1
EdExcel Decision Mathematics 1 Linear Programming Section 1: Formulating and solving graphically Notes and Examples These notes contain subsections on: Formulating LP problems Solving LP problems Minimisation
More informationLinear Programming Supplement E
Linear Programming Supplement E Linear Programming Linear programming: A technique that is useful for allocating scarce resources among competing demands. Objective function: An expression in linear programming
More informationLinear Programming Notes V Problem Transformations
Linear Programming Notes V Problem Transformations 1 Introduction Any linear programming problem can be rewritten in either of two standard forms. In the first form, the objective is to maximize, the material
More informationThis activity will show you how to draw graphs of algebraic functions in Excel.
This activity will show you how to draw graphs of algebraic functions in Excel. Open a new Excel workbook. This is Excel in Office 2007. You may not have used this version before but it is very much the
More informationUniversity of Southern California Marshall Information Services
University of Southern California Marshall Information Services Determine Breakeven Price Using Excel - Using Goal Seek, Data Tables, Vlookup & Charts This guide covers how to determine breakeven price
More informationImages of Microsoft Excel dialog boxes Microsoft. All rights reserved. This content is excluded from our Creative Commons license.
1 Images of Microsoft Excel dialog boxes Microsoft. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/. Tool
More informationUsing EXCEL Solver October, 2000
Using EXCEL Solver October, 2000 2 The Solver option in EXCEL may be used to solve linear and nonlinear optimization problems. Integer restrictions may be placed on the decision variables. Solver may be
More informationEXCEL SOLVER TUTORIAL
ENGR62/MS&E111 Autumn 2003 2004 Prof. Ben Van Roy October 1, 2003 EXCEL SOLVER TUTORIAL This tutorial will introduce you to some essential features of Excel and its plug-in, Solver, that we will be using
More informationOptimal Scheduling for Dependent Details Processing Using MS Excel Solver
BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume 8, No 2 Sofia 2008 Optimal Scheduling for Dependent Details Processing Using MS Excel Solver Daniela Borissova Institute of
More informationLinear Programming. March 14, 2014
Linear Programming March 1, 01 Parts of this introduction to linear programming were adapted from Chapter 9 of Introduction to Algorithms, Second Edition, by Cormen, Leiserson, Rivest and Stein [1]. 1
More informationThe Graphical Method: An Example
The Graphical Method: An Example Consider the following linear program: Maximize 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2 0, where, for ease of reference,
More information3. Evaluate the objective function at each vertex. Put the vertices into a table: Vertex P=3x+2y (0, 0) 0 min (0, 5) 10 (15, 0) 45 (12, 2) 40 Max
SOLUTION OF LINEAR PROGRAMMING PROBLEMS THEOREM 1 If a linear programming problem has a solution, then it must occur at a vertex, or corner point, of the feasible set, S, associated with the problem. Furthermore,
More informationSolving Linear Programs using Microsoft EXCEL Solver
Solving Linear Programs using Microsoft EXCEL Solver By Andrew J. Mason, University of Auckland To illustrate how we can use Microsoft EXCEL to solve linear programming problems, consider the following
More informationExcel Tutorial. Bio 150B Excel Tutorial 1
Bio 15B Excel Tutorial 1 Excel Tutorial As part of your laboratory write-ups and reports during this semester you will be required to collect and present data in an appropriate format. To organize and
More informationQuestion 2: How will changes in the objective function s coefficients change the optimal solution?
Question 2: How will changes in the objective function s coefficients change the optimal solution? In the previous question, we examined how changing the constants in the constraints changed the optimal
More informationCHAPTER 11: BASIC LINEAR PROGRAMMING CONCEPTS
Linear programming is a mathematical technique for finding optimal solutions to problems that can be expressed using linear equations and inequalities. If a real-world problem can be represented accurately
More informationSpecial Situations in the Simplex Algorithm
Special Situations in the Simplex Algorithm Degeneracy Consider the linear program: Maximize 2x 1 +x 2 Subject to: 4x 1 +3x 2 12 (1) 4x 1 +x 2 8 (2) 4x 1 +2x 2 8 (3) x 1, x 2 0. We will first apply the
More informationLinear Programming Notes VII Sensitivity Analysis
Linear Programming Notes VII Sensitivity Analysis 1 Introduction When you use a mathematical model to describe reality you must make approximations. The world is more complicated than the kinds of optimization
More informationFREE FALL. Introduction. Reference Young and Freedman, University Physics, 12 th Edition: Chapter 2, section 2.5
Physics 161 FREE FALL Introduction This experiment is designed to study the motion of an object that is accelerated by the force of gravity. It also serves as an introduction to the data analysis capabilities
More informationA Guide to Using Excel in Physics Lab
A Guide to Using Excel in Physics Lab Excel has the potential to be a very useful program that will save you lots of time. Excel is especially useful for making repetitious calculations on large data sets.
More informationOPRE 6201 : 2. Simplex Method
OPRE 6201 : 2. Simplex Method 1 The Graphical Method: An Example Consider the following linear program: Max 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2
More informationLecture 3. Linear Programming. 3B1B Optimization Michaelmas 2015 A. Zisserman. Extreme solutions. Simplex method. Interior point method
Lecture 3 3B1B Optimization Michaelmas 2015 A. Zisserman Linear Programming Extreme solutions Simplex method Interior point method Integer programming and relaxation The Optimization Tree Linear Programming
More informationMS Excel. Handout: Level 2. elearning Department. Copyright 2016 CMS e-learning Department. All Rights Reserved. Page 1 of 11
MS Excel Handout: Level 2 elearning Department 2016 Page 1 of 11 Contents Excel Environment:... 3 To create a new blank workbook:...3 To insert text:...4 Cell addresses:...4 To save the workbook:... 5
More informationAirport Planning and Design. Excel Solver
Airport Planning and Design Excel Solver Dr. Antonio A. Trani Professor of Civil and Environmental Engineering Virginia Polytechnic Institute and State University Blacksburg, Virginia Spring 2012 1 of
More informationFinite Mathematics Using Microsoft Excel
Overview and examples from Finite Mathematics Using Microsoft Excel Revathi Narasimhan Saint Peter's College An electronic supplement to Finite Mathematics and Its Applications, 6th Ed., by Goldstein,
More informationEXCEL Tutorial: How to use EXCEL for Graphs and Calculations.
EXCEL Tutorial: How to use EXCEL for Graphs and Calculations. Excel is powerful tool and can make your life easier if you are proficient in using it. You will need to use Excel to complete most of your
More informationUsing Excel s Solver
Using Excel s Solver Contents Page Answer Complex What-If Questions Using Microsoft Excel Solver........... 1 When to Use Solver Identifying Key Cells in Your Worksheet Solver Settings are Persistent Saving
More informationEngineering Problem Solving and Excel. EGN 1006 Introduction to Engineering
Engineering Problem Solving and Excel EGN 1006 Introduction to Engineering Mathematical Solution Procedures Commonly Used in Engineering Analysis Data Analysis Techniques (Statistics) Curve Fitting techniques
More informationOperation Research. Module 1. Module 2. Unit 1. Unit 2. Unit 3. Unit 1
Operation Research Module 1 Unit 1 1.1 Origin of Operations Research 1.2 Concept and Definition of OR 1.3 Characteristics of OR 1.4 Applications of OR 1.5 Phases of OR Unit 2 2.1 Introduction to Linear
More informationLinear Programming for Optimization. Mark A. Schulze, Ph.D. Perceptive Scientific Instruments, Inc.
1. Introduction Linear Programming for Optimization Mark A. Schulze, Ph.D. Perceptive Scientific Instruments, Inc. 1.1 Definition Linear programming is the name of a branch of applied mathematics that
More informationQuickstart for Desktop Version
Quickstart for Desktop Version What is GeoGebra? Dynamic Mathematics Software in one easy-to-use package For learning and teaching at all levels of education Joins interactive 2D and 3D geometry, algebra,
More informationExperiment 1A: Excel Graphing Exercise
Physics 210 Lab Manual Page 1-1 Experiment 1A: Excel Graphing Exercise The data for this exercise comes from measuring the time for a simple pendulum to swing back and forth 20 times as a function of the
More informationExcel 2007 Basic knowledge
Ribbon menu The Ribbon menu system with tabs for various Excel commands. This Ribbon system replaces the traditional menus used with Excel 2003. Above the Ribbon in the upper-left corner is the Microsoft
More informationIntroduction to Linear Programming (LP) Mathematical Programming (MP) Concept
Introduction to Linear Programming (LP) Mathematical Programming Concept LP Concept Standard Form Assumptions Consequences of Assumptions Solution Approach Solution Methods Typical Formulations Massachusetts
More informationStandard Form of a Linear Programming Problem
494 CHAPTER 9 LINEAR PROGRAMMING 9. THE SIMPLEX METHOD: MAXIMIZATION For linear programming problems involving two variables, the graphical solution method introduced in Section 9. is convenient. However,
More informationInteractive Excel Spreadsheets:
Interactive Excel Spreadsheets: Constructing Visualization Tools to Enhance Your Learner-centered Math and Science Classroom Scott A. Sinex Department of Physical Sciences and Engineering Prince George
More informationWhat is Linear Programming?
Chapter 1 What is Linear Programming? An optimization problem usually has three essential ingredients: a variable vector x consisting of a set of unknowns to be determined, an objective function of x to
More informationLinear Programming II: Minimization 2006 Samuel L. Baker Assignment 11 is on page 16.
LINEAR PROGRAMMING II 1 Linear Programming II: Minimization 2006 Samuel L. Baker Assignment 11 is on page 16. Introduction A minimization problem minimizes the value of the objective function rather than
More informationTutorial on Using Excel Solver to Analyze Spin-Lattice Relaxation Time Data
Tutorial on Using Excel Solver to Analyze Spin-Lattice Relaxation Time Data In the measurement of the Spin-Lattice Relaxation time T 1, a 180 o pulse is followed after a delay time of t with a 90 o pulse,
More informationHow To Run Statistical Tests in Excel
How To Run Statistical Tests in Excel Microsoft Excel is your best tool for storing and manipulating data, calculating basic descriptive statistics such as means and standard deviations, and conducting
More informationWhat does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of y = mx + b.
PRIMARY CONTENT MODULE Algebra - Linear Equations & Inequalities T-37/H-37 What does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of
More information4.6 Linear Programming duality
4.6 Linear Programming duality To any minimization (maximization) LP we can associate a closely related maximization (minimization) LP. Different spaces and objective functions but in general same optimal
More informationMBA Quantitative Methods PC-Exercises Introductory Examples
MBA Quantitative Methods PC-Exercises Introductory Examples intro.xls intro_with_output.xls intro.doc For all Examples you need the file intro.xls. The file intro_with_output.xls is the file with the results
More informationUsing the Simplex Method to Solve Linear Programming Maximization Problems J. Reeb and S. Leavengood
PERFORMANCE EXCELLENCE IN THE WOOD PRODUCTS INDUSTRY EM 8720-E October 1998 $3.00 Using the Simplex Method to Solve Linear Programming Maximization Problems J. Reeb and S. Leavengood A key problem faced
More informationMODELLING. IF...THEN Function EXCEL 2007. Wherever you see this symbol, make sure you remember to save your work!
MODELLING IF THEN IF...THEN Function EXCEL 2007 Wherever you see this symbol, make sure you remember to save your work! IF.Then Function Some functions do not calculate values but instead do logical tests
More informationIEOR 4404 Homework #2 Intro OR: Deterministic Models February 14, 2011 Prof. Jay Sethuraman Page 1 of 5. Homework #2
IEOR 4404 Homework # Intro OR: Deterministic Models February 14, 011 Prof. Jay Sethuraman Page 1 of 5 Homework #.1 (a) What is the optimal solution of this problem? Let us consider that x 1, x and x 3
More informationOptimization Modeling for Mining Engineers
Optimization Modeling for Mining Engineers Alexandra M. Newman Division of Economics and Business Slide 1 Colorado School of Mines Seminar Outline Linear Programming Integer Linear Programming Slide 2
More informationMicrosoft Excel Basics
COMMUNITY TECHNICAL SUPPORT Microsoft Excel Basics Introduction to Excel Click on the program icon in Launcher or the Microsoft Office Shortcut Bar. A worksheet is a grid, made up of columns, which are
More informationLinear Programming I
Linear Programming I November 30, 2003 1 Introduction In the VCR/guns/nuclear bombs/napkins/star wars/professors/butter/mice problem, the benevolent dictator, Bigus Piguinus, of south Antarctica penguins
More informationLinear programming. Learning objectives. Theory in action
2 Linear programming Learning objectives After finishing this chapter, you should be able to: formulate a linear programming model for a given problem; solve a linear programming model with two decision
More informationDealing with Data in Excel 2010
Dealing with Data in Excel 2010 Excel provides the ability to do computations and graphing of data. Here we provide the basics and some advanced capabilities available in Excel that are useful for dealing
More informationQuestion 2: How do you solve a linear programming problem with a graph?
Question 2: How do you solve a linear programming problem with a graph? Now that we have several linear programming problems, let s look at how we can solve them using the graph of the system of inequalities.
More information1 Solving LPs: The Simplex Algorithm of George Dantzig
Solving LPs: The Simplex Algorithm of George Dantzig. Simplex Pivoting: Dictionary Format We illustrate a general solution procedure, called the simplex algorithm, by implementing it on a very simple example.
More informationUsing Formulas, Functions, and Data Analysis Tools Excel 2010 Tutorial
Using Formulas, Functions, and Data Analysis Tools Excel 2010 Tutorial Excel file for use with this tutorial Tutor1Data.xlsx File Location http://faculty.ung.edu/kmelton/data/tutor1data.xlsx Introduction:
More informationBasic Excel Handbook
2 5 2 7 1 1 0 4 3 9 8 1 Basic Excel Handbook Version 3.6 May 6, 2008 Contents Contents... 1 Part I: Background Information...3 About This Handbook... 4 Excel Terminology... 5 Excel Terminology (cont.)...
More informationUsing Excel (Microsoft Office 2007 Version) for Graphical Analysis of Data
Using Excel (Microsoft Office 2007 Version) for Graphical Analysis of Data Introduction In several upcoming labs, a primary goal will be to determine the mathematical relationship between two variable
More informationScientific Graphing in Excel 2010
Scientific Graphing in Excel 2010 When you start Excel, you will see the screen below. Various parts of the display are labelled in red, with arrows, to define the terms used in the remainder of this overview.
More informationUsing Microsoft Excel to Plot and Analyze Kinetic Data
Entering and Formatting Data Using Microsoft Excel to Plot and Analyze Kinetic Data Open Excel. Set up the spreadsheet page (Sheet 1) so that anyone who reads it will understand the page (Figure 1). Type
More informationwith functions, expressions and equations which follow in units 3 and 4.
Grade 8 Overview View unit yearlong overview here The unit design was created in line with the areas of focus for grade 8 Mathematics as identified by the Common Core State Standards and the PARCC Model
More informationYears after 2000. US Student to Teacher Ratio 0 16.048 1 15.893 2 15.900 3 15.900 4 15.800 5 15.657 6 15.540
To complete this technology assignment, you should already have created a scatter plot for your data on your calculator and/or in Excel. You could do this with any two columns of data, but for demonstration
More informationSpreadsheets and Laboratory Data Analysis: Excel 2003 Version (Excel 2007 is only slightly different)
Spreadsheets and Laboratory Data Analysis: Excel 2003 Version (Excel 2007 is only slightly different) Spreadsheets are computer programs that allow the user to enter and manipulate numbers. They are capable
More informationALGEBRA. sequence, term, nth term, consecutive, rule, relationship, generate, predict, continue increase, decrease finite, infinite
ALGEBRA Pupils should be taught to: Generate and describe sequences As outcomes, Year 7 pupils should, for example: Use, read and write, spelling correctly: sequence, term, nth term, consecutive, rule,
More informationExcel Math Project for 8th Grade Identifying Patterns
There are several terms that we will use to describe your spreadsheet: Workbook, worksheet, row, column, cell, cursor, name box, formula bar. Today you are going to create a spreadsheet to investigate
More informationObjectives. Day Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7. Inventory 2,782 2,525 2,303 2,109 1,955 1,788 1,570
Activity 4 Objectives Use the CellSheet App to predict future trends based on average daily use Use a linear regression model to predict future trends Introduction Have you ever gone to buy a new CD or
More informationExcel Level Two. Introduction. Contents. Exploring Formulas. Entering Formulas
Introduction Excel Level Two This workshop introduces you to formulas, functions, moving and copying data, using autofill, relative and absolute references, and formatting cells. Contents Introduction
More informationNotes on Excel Forecasting Tools. Data Table, Scenario Manager, Goal Seek, & Solver
Notes on Excel Forecasting Tools Data Table, Scenario Manager, Goal Seek, & Solver 2001-2002 1 Contents Overview...1 Data Table Scenario Manager Goal Seek Solver Examples Data Table...2 Scenario Manager...8
More informationIntroduction to Microsoft Excel 2007/2010
to Microsoft Excel 2007/2010 Abstract: Microsoft Excel is one of the most powerful and widely used spreadsheet applications available today. Excel's functionality and popularity have made it an essential
More informationGraphing Parabolas With Microsoft Excel
Graphing Parabolas With Microsoft Excel Mr. Clausen Algebra 2 California State Standard for Algebra 2 #10.0: Students graph quadratic functions and determine the maxima, minima, and zeros of the function.
More informationSensitivity Report in Excel
The Answer Report contains the original guess for the solution and the final value of the solution as well as the objective function values for the original guess and final value. The report also indicates
More informationEducator s Guide to Excel Graphing
Educator s Guide to Excel Graphing Overview: Students will use Excel to enter data into a spreadsheet and make a graph. Grades and Subject Areas: Grade 3-6 Subject Math Objectives: Students will: make
More informationSummary of important mathematical operations and formulas (from first tutorial):
EXCEL Intermediate Tutorial Summary of important mathematical operations and formulas (from first tutorial): Operation Key Addition + Subtraction - Multiplication * Division / Exponential ^ To enter a
More informationTIPS FOR DOING STATISTICS IN EXCEL
TIPS FOR DOING STATISTICS IN EXCEL Before you begin, make sure that you have the DATA ANALYSIS pack running on your machine. It comes with Excel. Here s how to check if you have it, and what to do if you
More informationEfficient Portfolios in Excel Using the Solver and Matrix Algebra
Efficient Portfolios in Excel Using the Solver and Matrix Algebra This note outlines how to use the solver and matrix algebra in Excel to compute efficient portfolios. The example used in this note is
More informationLinear Programming. April 12, 2005
Linear Programming April 1, 005 Parts of this were adapted from Chapter 9 of i Introduction to Algorithms (Second Edition) /i by Cormen, Leiserson, Rivest and Stein. 1 What is linear programming? The first
More informationThe Center for Teaching, Learning, & Technology
The Center for Teaching, Learning, & Technology Instructional Technology Workshops Microsoft Excel 2010 Formulas and Charts Albert Robinson / Delwar Sayeed Faculty and Staff Development Programs Colston
More informationLecture 2: August 29. Linear Programming (part I)
10-725: Convex Optimization Fall 2013 Lecture 2: August 29 Lecturer: Barnabás Póczos Scribes: Samrachana Adhikari, Mattia Ciollaro, Fabrizio Lecci Note: LaTeX template courtesy of UC Berkeley EECS dept.
More informationSOLVING EQUATIONS WITH EXCEL
SOLVING EQUATIONS WITH EXCEL Excel and Lotus software are equipped with functions that allow the user to identify the root of an equation. By root, we mean the values of x such that a given equation cancels
More informationHow To Analyze Data In Excel 2003 With A Powerpoint 3.5
Microsoft Excel 2003 Data Analysis Larry F. Vint, Ph.D lvint@niu.edu 815-753-8053 Technical Advisory Group Customer Support Services Northern Illinois University 120 Swen Parson Hall DeKalb, IL 60115 Copyright
More informationBelow is a very brief tutorial on the basic capabilities of Excel. Refer to the Excel help files for more information.
Excel Tutorial Below is a very brief tutorial on the basic capabilities of Excel. Refer to the Excel help files for more information. Working with Data Entering and Formatting Data Before entering data
More informationSpreadsheet - Introduction
CSCA0102 IT and Business Applications Chapter 6 Spreadsheet - Introduction Spreadsheet A spreadsheet (or spreadsheet program) is software that permits numerical data to be used and to perform automatic
More informationCalc Guide Chapter 9 Data Analysis
Calc Guide Chapter 9 Data Analysis Using Scenarios, Goal Seek, Solver, others Copyright This document is Copyright 2007 2011 by its contributors as listed below. You may distribute it and/or modify it
More informationSection 7.2 Linear Programming: The Graphical Method
Section 7.2 Linear Programming: The Graphical Method Man problems in business, science, and economics involve finding the optimal value of a function (for instance, the maimum value of the profit function
More informationSensitivity Analysis with Excel
Sensitivity Analysis with Excel 1 Lecture Outline Sensitivity Analysis Effects on the Objective Function Value (OFV): Changing the Values of Decision Variables Looking at the Variation in OFV: Excel One-
More informationExcel 2010: Create your first spreadsheet
Excel 2010: Create your first spreadsheet Goals: After completing this course you will be able to: Create a new spreadsheet. Add, subtract, multiply, and divide in a spreadsheet. Enter and format column
More information1. Graphing Linear Inequalities
Notation. CHAPTER 4 Linear Programming 1. Graphing Linear Inequalities x apple y means x is less than or equal to y. x y means x is greater than or equal to y. x < y means x is less than y. x > y means
More informationCreating a Gradebook in Excel
Creating a Spreadsheet Gradebook 1 Creating a Gradebook in Excel Spreadsheets are a great tool for creating gradebooks. With a little bit of work, you can create a customized gradebook that will provide
More informationCurve Fitting in Microsoft Excel By William Lee
Curve Fitting in Microsoft Excel By William Lee This document is here to guide you through the steps needed to do curve fitting in Microsoft Excel using the least-squares method. In mathematical equations
More informationBasic Components of an LP:
1 Linear Programming Optimization is an important and fascinating area of management science and operations research. It helps to do less work, but gain more. Linear programming (LP) is a central topic
More informationTutorial: Using Excel for Linear Optimization Problems
Tutorial: Using Excel for Linear Optimization Problems Part 1: Organize Your Information There are three categories of information needed for solving an optimization problem in Excel: an Objective Function,
More informationUsing Excel for Statistical Analysis
2010 Using Excel for Statistical Analysis Microsoft Excel is spreadsheet software that is used to store information in columns and rows, which can then be organized and/or processed. Excel is a powerful
More informationTo launch the Microsoft Excel program, locate the Microsoft Excel icon, and double click.
EDIT202 Spreadsheet Lab Assignment Guidelines Getting Started 1. For this lab you will modify a sample spreadsheet file named Starter- Spreadsheet.xls which is available for download from the Spreadsheet
More informationCreating, Solving, and Graphing Systems of Linear Equations and Linear Inequalities
Algebra 1, Quarter 2, Unit 2.1 Creating, Solving, and Graphing Systems of Linear Equations and Linear Inequalities Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned
More informationAlgebra Bridge Project Cell Phone Plans
Algebra Bridge Project Cell Phone Plans Name Teacher Part I: Two Cell Phone Plans You are in the market for a new cell phone, and you have narrowed your search to two different cell phone companies --
More informationE x c e l 2 0 1 0 : Data Analysis Tools Student Manual
E x c e l 2 0 1 0 : Data Analysis Tools Student Manual Excel 2010: Data Analysis Tools Chief Executive Officer, Axzo Press: Series Designer and COO: Vice President, Operations: Director of Publishing Systems
More information