1. LINEAR PROGRAMMING INTRODUCTION
Chapter 7 of Quantitative Methods for Business by Anderson, Sweeney and Williams
Read sections 7.1, 7.2, 7.3, 7.4, 7.5, 7.6, 7.7 and appendix 7.1

2. LINEAR PROGRAMMING One of the most widely used M.S. tools
Used to solve a wide variety of problems
Human Resource Scheduling
Routing of Delivery Vehicles
Selection of Advertising Media
Planning Product Levels in Manufacturing
Blending in Oil Refineries
Selection of Investment Opportunities
and many, many more

3. LINEAR PROGRAMMING
Most widely used solution technique: The Simplex Method
Developed by George Dantzig in the 1940’s
First used to solve military operations problems during WWII
Coincided with the development of the first widely used computers

4. LINEAR PROGRAMMING MODELS
Objective is to minimize or maximize some linear function of the decision variables
Constraints are linear equations or inequalities

5. Homework status
Before the next class, you should complete the following homework problems in chapter 7 #1

6. LINEAR PROGRAMMING MODELS
Extremely large models easily solvable using existing computer programs
Consider using L.P. Models whenever you are faced with allocating scarce resources among competing activities

7. LINEAR PROGRAMMING--Introduction to Formulation
3 Steps to Formulate A Linear Programming Problem:
DEFINE DECISION VARIABLES
FORMULATE OBJECTIVE
IDENTIFY CONSTRAINTS

8. Glass Manufacturer Example
A glass company is considering using its excess capacity to manufacture two new products:
wood frame windows, which earn $3 profit each, and
aluminum frame doors which earn $5 profit, each.

9. Glass Manufacturer Example
There are
4 man-hours / day available in plant 1 (Plant 1 is used for wood frames)
12 man-hours / day available in plant 2 (Plant 2 for aluminum frames)
18 man-hours / day available in plant 3 (Plant 3 for glass and assembly)

10. Glass Manufacturer Example
Each unit of the new products would require the following plant resources (in man-hours):
WINDOWS DOORS
PLANT
PLANT
PLANT

11. Glass Manufacturer Example
Formulate the problem:
1. DEFINE DECISION VARIABLES

12. Glass Manufacturer Example
2. OBJECTIVE FUNCTION

13. Glass Manufacturer Example
3. IDENTIFY CONSTRAINTS

14. GRAPHICAL SOLUTION TECHNIQUE
Associate the horizontal axis with one decision variable, and associate the vertical axis with the other.
Draw the constraints.
Identify the feasible region as the area where all the constraints intersect (are “satisfied”).

15. GRAPHICAL SOLUTION TECHNIQUE
Find the optimal solution (the feasible point which gives the best value of the objective).
Graph the objective function line for any 2 arbitrary values of Z
Identify the improving direction for Z
Move a pen parallel to the Z lines in the improving direction as far as possible
Last feasible point pen touches is the optimal solution

16. GRAPHICAL SOLUTION TECHNIQUE
How to find the coordinates of the optimal point?
Identify the (2) constraints which go through the point
Solve those (2) constraint equations simultaneously

17. Homework status
Before the next class, you should complete the following homework problems in chapter 7
#3, #7, #8, #11 (by hand), #24 (by hand), and #31 (by hand)

18. POSSIBLE ANSWERS A unique (extreme point) optimal solution
Alternate optimal solutions
An unbounded solution
An infeasible problem

19. IMPORTANT ASSUMPTIONS OF LINEAR PROGRAMMING PROBLEMS
LINEARITY OF OBJECTIVE AND CONSTRAINTS
i.e. they can be written in the form:
C1X1+C2X2+…+CnXn  RHS
where the Xis are the decision variables, the Cis are (constant) coefficients, and RHS is the (constant) RHS
DIVISIBILITY OF DECISION VARIABLES
i.e. they may take fractional values

20. Using Excel to solve an LP
Label one row for each of these:
Objective function coefficients
Values of the decision variables
Each constraint.
Label one column for each of these:
Each decision variable
Total value of the left hand side
Right hand side value.

21. Using Excel to solve an LP
Key in coefficients from constraints (blue border)
Key in right-hand-side values for constraints (blue border)
Key in objective function coefficients (blue border)

22. Using Excel to solve an LP
Designate cells for decision values (red border)
Designate cell for objective function value (double black border)

23. Using Excel to solve an LP
Specify formula for left-hand-sides of constraints, using the SUMPRODUCT function
Specify formula of Objective Function Value cell, using the SUMPRODUCT function

24. Using Excel to solve an LP
(Key in any values for decision variables
Try different values, to see what happens to left-hand- side values and objective value)
Click Data  Analysis  Solver
Specify “Target Cell” as the objective function value (cell with double black border)

25. Using Excel to solve an LP
Click MAX (or MIN)
Specify “changing cells” as decision variable value cells (cells with red border)
Click “Add” to add constraints, one (or several) at a time by selecting each:
Total Left-hand-side cell(s)
Appropriate inequality/equality sign
Right-hand-side cell(s)

26. Using Excel to solve an LP
Be sure the box next to “Make Unconstrained Variables Non-negative” is checked.
By “Select a Solving Method”, choose “Simplex LP”
Click SOLVE
Study the solution to be sure it is reasonable (including feasible) and modify/correct the model and resolve if it is not.

27. Homework help
See Solver Hints file (on Blackboard, under Module2 Linear Programming) if you need help.

28. Homework status
Before the next class, you should complete the following homework problems in chapter 7
#11 (on Excel), #24 (on Excel), #31 (on Excel)