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) 1 4
Plant 2 Usage (Hours) 2 12
Plant 3 Usage (Hours) 3 18
Unit Profit
$3,000
$5,000

6. Mathematical Model Let x1 = number of doors produced per week
Let x2 = number of windows produced per week
Let Z = profit per week
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑍=3 𝑥 1 +5 𝑥 2
Subject to:
𝑥 1 ≤4
2 𝑥 2 ≤12
3𝑥 1 + 2𝑥 2 ≤18
𝑥 1 ≥0, 𝑥 2 ≥0

7. Graphical Solution x1=4 x2 9 2x2=12 6 3x1+2x2=18 Z=3x1+5x2=364 6 x1
Z=3x1+5x2=10

8. The General Linear Programming Model
Z = measure of performance
xj = 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

9. 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

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 ≤𝑏1
𝑎 21 𝑥 1 + 𝑎 22 𝑥 2 +… +𝑐 2𝑛 𝑥 𝑛 ≤𝑏2 .
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 xj could be unrestricted in sign, i.e., xj < 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 3.1-10 in the textbook Develop a mathematical model
Solve the problem using the graphical method
Solve the problem using excel by developing a spreadsheet model

22. In-Class Activity (Not Graded)
Solve the following using the graphical method and the spreadsheet method:
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑍=200 𝑥 𝑥 2
Subject to:
2 0𝑥 1 −1 0𝑥 2 ≤150
15𝑥 𝑥 2 ≤180
3 𝑥 1 +5 𝑥 2 ≤45
𝑥 1 ≥0, 𝑥 2 ≥0