Introduction to Linear ProgrammingChapter 3: Hillier and LiebermanChapter 3: Decision Tools for AgribusinessDr. Hurley’s AGB 328 Course

Terms to KnowSimplex 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

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

Wyndor Glass Co. ExampleCompany has two new products

—a door and a windowThe company has three plants to

develop these two new productsThe goal of the company is to

maximize profits

Key Data for Wyndor Doors Window

sTime Available Hours

Plant 1 Usage (Hours)

1 0 4

Plant 2 Usage (Hours)

0 2 12

Plant 3 Usage (Hours)

3 2 18

Unit Profit $3,000 $5,000

Mathematical ModelLet x1 = number of doors produced per

weekLet x2 = number of windows produced

per weekLet Z = profit per week

Subject to:

2

Graphical Solution

x2

0 6

9

6

4

Z=3x1+5x2=36

3x1+2x2=18

2x2=12

x1

x1=4

Z=3x1+5x2=20

Z=3x1+5x2=10

The General Linear Programming ModelZ = measure of performancexj = a decision variable that indicates how

much you are doing of activity j for j = 1, 2, …, n

cj = a parameter that converts activity j into the overall measure of performance

bi = the amount of resource i you have available to allocate to your different activities for i = 1, 2, …, m

aij = a parameter that converts activity j into the amount of resource i used

Resource Allocation Data Matrix

Activity 1

Activity 2

… Activity n

Resource Available

Resource 1 a11 a12 … a1n b1

Resource 2 a21 a22 … a2n b2

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.Resource m am1 am2 … amn bm

Contribution to Z

c1 c2 … cn

Standard Mathematical FormSubject to:

. .

, …,

Changes that Can Be Made to the Standard Form The objective function could be

minimized instead of maximizedThe functional constraints can be

met with equality (=) or greater than (≥) signs

The decision variables xj could be unrestricted in sign, i.e., xj < 0 is also possible

Major Assumptions Behind Linear ProgrammingAll functions are linearProportionality AssumptionAdditivityDivisibilityCertainty

Solving Linear Programming Problems Using a SpreadsheetExcel 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:

Guidelines for Building Good SpreadsheetsEnter 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.

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.

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.

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.

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.

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.

Review the Following Spreadsheet ModelsWyndor GlassRadiation TherapyKibbutzimNori and LeetsSave-ItUnion AirwaysDistribution Unlimited

Example3.1-10 in the textbook

◦Develop a mathematical model◦Solve the problem using the

graphical method◦Solve the problem using excel by

developing a spreadsheet model

In-Class Activity (Not Graded)Solve the following using the

graphical method and the spreadsheet method:

Subject to: