Introduction to Linear Programming. Chapter 3: Hillier and Lieberman Chapter 3: Decision Tools for Agribusiness Dr. Hurley s AGB 328 Course

Transcription

1 Introduction to Linear Programming Chapter 3: Hillier and Lieberman Chapter 3: Decision Tools for Agribusiness Dr Hurley s AGB 328 Course

2 Terms to Know Simplex Method, Feasible Region, Slope- Intercept Form, Optimal Solution, Graphical Method, Decision Variables, Parameters, Objective Function, Constraints, Functional Constraints, Non- Negativity Constraints, Feasible Solution, Infeasible Solution, Feasible Region

3 Terms to Know Cont No Feasible Solution, Optimal Solution, Most Favorable Value, Multiple Optimal Solutions, No Optimal Solutions, Unbounded Z, Corner-Point Feasible Solution (CPF), Blending Problem, Data Cells, Range Name, Changing Cells, Output Cells, Target Cell

4 Wyndor Glass Co Example Company has two new products a door and a window The company has three plants to develop these two new products The goal of the company is to maximize profits

5 Key Data for Wyndor Doors Windows Time Available Hours Plant 1 Usage (Hours) Plant 2 Usage (Hours) Plant 3 Usage (Hours) Unit Profit $3,000 $5,000

6 Mathematical Model Let x 1 = number of doors produced per week Let x 2 = number of windows produced per week Let Z = profit per week Maximize Z = 3x 1 + 5x 2 Subject to: x 1 4 2x x 1 + 2x 2 18 x 1 0, x 2 0

7 Graphical Solution x 2 x 1 = x 2 =12 3x 1 +2x 2 =18 Z=3x 1 +5x 2 = Z=3x 1 +5x 2 =20 x 1 Z=3x 1 +5x 2 =10

8 The General Linear Programming Model Z = measure of performance x j = a decision variable that indicates how much you are doing of activity j for j = 1, 2,, n c j = a parameter that converts activity j into the overall measure of performance b i = the amount of resource i you have available to allocate to your different activities for i = 1, 2,, m a ij = a parameter that converts activity j into the amount of resource i used

9 Resource Allocation Data Matrix Activity 1 Activity 2 Activity n Resource Available Resource 1 a 11 a 12 a 1n b 1 Resource 2 a 21 a 22 a 2n b 2 Resource m a m1 a m2 a mn b m Contribution to Z c 1 c 2 c n

10 Standard Mathematical Form Maximize Z = c 1 x 1 + c 2 x 2 + +c n x n Subject to: a 11 x 1 + a 12 x 2 + +c 1n x n b 1 a 21 x 1 + a 22 x 2 + +c 2n x n b 2 a m1 x 1 + a m2 x 2 + +c mn x n b m x 1 0, x 2 0,, x n 0

11 Changes that Can Be Made to the Standard Form The objective function could be minimized instead of maximized The functional constraints can be met with equality (=) or greater than ( ) signs The decision variables x j could be unrestricted in sign, ie, x j < 0 is also possible

12 Major Assumptions Behind Linear Programming All functions are linear Proportionality Assumption Additivity Divisibility Certainty

13 Solving Linear Programming Problems Using a Spreadsheet Excel has an add-in called Solver that can solve linear programming problems Major components to Solver are: Set Objective: To: By Changing Variable Cells: Subject to the Constraints: Make Unconstrained Variables Non-negative should be checked Select a Solving Method:

14 Guidelines for Building Good Spreadsheets Enter the data first Since the data can dictate the structure of the spreadsheet model, it is valuable to input the data in the spreadsheet first This can also allow you to build the spreadsheet to closely resemble the structure of the data

15 Guidelines for Building Good Spreadsheets Cont Organize and clearly identify the data Data should be grouped together in a convenient format Each piece of data or group of data should be appropriately labeled Enter each piece of data into one cell only If you need to use the data elsewhere in the model, you can reference it

16 Guidelines for Building Good Spreadsheets Cont Separate data from formulas If possible, formulas should have no specific parameters encoded in them By keeping the data separate from formulas, you can save time when changes are needed by only having to change one parameter rather than looking for all the formulas that use a specific piece of data This allows all the data to be visual in the spreadsheet

17 Guidelines for Building Good Spreadsheets Cont Keep it simple You should avoid more powerful functions when simpler ones will accomplish the same task Keep formulas simple If you have a very complicated formula, you should break it up into components on the spreadsheet Use range names Range names should be indicative of what they represent When using range names, care should be taken not to allow too many range names so the names become unwieldy

18 Guidelines for Building Good Spreadsheets Cont Use relative and absolute references to simplify copying formulas This also allows you to copy cells without making as many errors Use borders, shading, and colors to distinguish between cell types This will make it easy for you to keep track of the items within your spreadsheet model

19 Guidelines for Building Good Spreadsheets Cont Show the entire model on the spreadsheet You should attempt to put as many of the elements of the model on the spreadsheet This will allow others to more easily understand your model This will allow people using the spreadsheet to more easily understand the Solver dialog box

20 Review the Following Spreadsheet Models Wyndor Glass Radiation Therapy Kibbutzim Nori and Leets Save-It Union Airways Distribution Unlimited

21 Example in the textbook Develop a mathematical model Solve the problem using the graphical method Solve the problem using excel by developing a spreadsheet model

22 In-Class Activity (Not Graded) Solve the following using the graphical method and the spreadsheet method: Maximize Z = 200x x 2 Subject to: 20x 1 10x x x x 1 + 5x 2 45 x 1 0, x 2 0