1. QM B Linear Programming

2. Overview What is linear programming (LP)? Formulating LPs
The Stratton company
Graphical insight
Using Excel Solver to solve LPs
Mile-High Microbrewery

3. What is linear programming?
It is NOT computer programming. Programming here means planning.
Mathematical technique
Optimization technique
A decision needs to made
Our goal is to determine the ‘best’ or ‘optimum’ decision
There are scarce resources available and/or specified requirements for achieving our goal.

4. Proctor and Gamble North American Product Sourcing
60 plants
10 distribution centers
1000 customer zones
Save $200 million dollars annually
Which products should be produced in which plants?
Which plants should supply which distribution centers?
Which distribution centers should supply which customer zones.

5. What does an LP look like?
There is a goal: objective function
Maximized or minimized
Written as a linear equation
There are scarce resources, restrictions and/or requirements: constraints
Limits your ability to achieve the goal

6. Formulating an LP: Stratton Co.
Produces two basic types of plastic pipes
Three resources have been identified as critical to pipe output
Pipe extrusion hours
Packaging hours
Special additive mix

7. Stratton Company Data Product Resource Availability Resource Type 1
Extrusion
4 hrs.
6 hrs.
48 hrs.
Packaging
2 hrs.
18 hrs.
Additive Mix
2 lbs.
1 lbs.
16 lbs.
Profit
$34
$40
All data given is for a package of pipe – 100 feet

8. Stratton Company (cont)
Formulate an LP model to determine how much of each type of pipe should be produced to maximize profit.

9. Three questions to formulate an LP:
What is the decision to be made?
Stratton Company
How much of pipe 1 to produce
How much of pipe 2 to produce
Defines the variables (if you are specific enough).
P1 – number of packages of Pipe 1 to produce
P2 – number of packages of Pipe 2 to produce

10. Question 2 for Formulating an LP:
What is the goal?
Stratton Company
Maximize profit
Defines the objective function
MAX 34 P P2

11. Question 3 for Formulating an LP:
What are the limited resources or requirements?
Extrusion hours
Packaging hours
Additive mix
4P1 + 6P2  48
2P1 + 2P2  18
2P1 + 1P2  16

12. LP for Stratton Company
Objective Function
LP for Stratton Company
MAX 34 P P2
Subject to:
4 P1 + 6 P2  48 Extrusion hours
2 P1 + 2 P2  18 Packaging hours
2 P1 + 1 P2  16 Additive supply
P1  0 and P2  0 Non-negativity
Constraints

13. Solving LPs ‘What if’ analysis (go to Excel) Graphical analysis
For insight
Simplex method
Solver – an Excel add-in
Computer packages designed for linear optimization

14. Graphical analysis – non-negativity

15. Graphical analysis – Extrusion constraint
4 P1 + 6 P2  48

16. Graphical analysis – Packaging Constraint
2 P1 + 2 P2  18

17. Graphical analysis – Additive supply constraint
Feasible Region
2 P1 + 1 P2  16

18. Which is the optimal solution?
Feasible Region
Set of all solutions that satisfy all of the constraints
Infinite number of solutions
Which is the optimal solution?
One of the solutions at the corner points

19. Corner point solutions
Feasible Region
MAX 34 P P2 (go to Excel)

20. Stratton Company – Summary
Optimal solution
P1 = 3
P2 = 6
Max = $342
The optimal product mix is 3 packages of Pipe 1 and 6 packages of Pipe 2. This provides a maximum profit of $342.

21. Setting up Excel Solver to solve LPs
Solver is an add-in to Excel
Not automatically ready
To get solver ready:
In Excel
Tools -> Add ins
Scroll down to Solver Add in
Check the box
Click on OK
Only need to do this one time

22. Mile-High Microbrewery
Mile-High Microbrewery makes a light beer and a dark beer. Mile-High has a limited supply of barley, limited bottling capacity, and a limited market for light beer. Profits are $0.20 per bottle of light beer and $0.50 per bottle of dark beer. Formulate an LP to maximize profits and determine how many bottles of each product should be produced per month.

23. Mile-High Microbrewery Data

24. Think-pair-share: Three questions:
What are the decisions to be made?
What is the goal?
What are the limited resources or requirements?

25. What are the decisions to be made?
L – Number of bottles of light beer to produce
D – Number of bottles of dark beer to produce

26. What is the goal?
Maximize profit
MAX L D

27. What are the limited resources or requirements?
Barley supply
0.10 L D  2000
Bottling capacity
1 L + 1 D  6000
Market capacity
1 L  4000

28. An aside for SUMPRODUCT function
2 groups of cells
Both in a row or both in a columns
Wish to multiply the corresponding entries then sum the products
=sumproduct(a2:c2, a3:c3)
= 2*5 + 3*6 + 4*7

29. Think-pair-share: SUMPRODUCT function
=SUMPRODUCT(B6:C6,B10:C10)
= 2*2 + 1*4 = 8

30. To solve an LP using Excel Solver
Setup the spreadsheet
TYPE data in one place (go to Excel)
CREATE Cells for decisions variables
ENTER formulas to calculate LHS of constraints
ENTER formulas to calculate Objective Function
Open solver box
Tools -> Solver

31. Excel Solver Dialog Box
Click on cell that calculates objective function
Select Max or Min
Click & drag to select decision variables
Click add to add the constraints

32. Excel solver – constraints dialog box
Select cell(s) with LHS
Select cell(s) with RHS
Select symbol (, , =)
Remember – Non-negativity constraints

33. Go to the Options Dialog box
Click options to assume linear model

34. Last dialog box - options
Click OK
Check the Assume linear models box
Check the Assume Non-negative box

35. Now SOLVE
Click solve to find optimal solution

36. Solver found a solution
Click on Answer and Sensitivity
Click OK

37. Answer and sensitivity reports