1. Introduction to Linear Programming
Professor Stephen Lawrence
Leeds School of Business
University of Colorado
Boulder, CO

2. Agenda Examples of LPs Graphical solution method Standard form
Assumptions of LP
Solving LP’s with Excel
More LP examples

3. An LP Example
Product Mix LP. A potter produces two products, a pitcher and a bowl. It takes about 1 hour to produce a bowl and requires 4 pounds of clay. A pitcher takes about 2 hours and consumes 3 pounds of clay. The profit on a bowl is $40 and $50 on a pitcher. She works 40 hours weekly, has 120 pounds of clay available each week, and wants more profits.

4. Another LP Example
“Diet” Problem. A farmer is preparing to plant a crop in the spring and needs to fertilize a field. There are two brands of fertilizer he can use: SuperGro and CropKwik. Each brand has a specific amount of nitrogen and phosphate. The field requires at least 16 pounds of nitrogen and 24 pounds of phosphate. SuperGro costs $6 per bag and CropKwik $3. How many bags of each type should the farmer use to adequately fertilize his field?
(lbs / bag)
Nitrogen
Phosphate
SuperGro 2 4
CropKwik 3

5. LP Marketing Example
Advertising Mix. Folly’s department store is working on an ad campaign for the summer using radio, TV, and newspaper ads, subject to the following information:
$100,000 budget
TV station has slots for 4 ads
Radio has slots for 10 ads
Newspaper has space for 7 ads
Ad agency has time/staff to produce no more than 15 ads
Exposure
Cost TV
20,000
$15,000
Radio
12,000
6,000
Print
9,000
4,000

6. Graphical LP Solutions
Works well for 2 decision variables
“Possible” for 3 decision variables
Impossible for 4+ variables
Other solution approaches necessary
Good to illustrate concepts, aid in conceptual understanding
An automated tool…

7. LP Standard Form Max Z = c1x1 + c2x2 + … + cnxn Subject to (s.t.)
a11x1 + a12x2 + … + a1nxn  b1 
a21x1 + a22x2 + … + a2nxn  b2 …
am1x1 + am2x2 + … + amnxn  bm
x1  0, x2  0, …, xn  0

8. LP Standard Form Max Z =  cj xj Subject to (s.t.)
 aij xj  bj i = 1, … , m
xj  0 j = 1, … , m
j =1 n

9. Assumptions of LP Linear objective function, constraints Divisibility
Proportionality
Additivity
Divisibility
Continuous decision variables
Certainty
Deterministic parameters

10. LP Concepts Decision variables Objective function Constraints
Feasible solutions
Feasible region (convex polytope)
Corner point solutions
Optimal solution
“Constrained optimization”

11. Solving LP’s with Excel Solver

12. Checking for Solver Standard with every version of Excel
Similar for QuattroPro and Lotus
Start up Excel
Look for “Solver…” in the Tools menu
If not there, you will need to add Solver from your original Excel distribution CD

13. Loading Solver Insert MS Office or Excel master CD
Click on “Tools | Add-Ins…”
Check “Solver Add-in” checkbox
Click “OK”
Solver will auto-load from CD
Solver now should appear in the “Tools” menu

14. Product Mix Example Max Z = 40x + 50y profits s.t. 1x + 2y  40 hours
Product Mix LP. A potter produces two products, a pitcher and a bowl. It takes about 1 hour to produce a bowl and requires 4 pounds of clay. A pitcher takes about 2 hours and consumes 3 pounds of clay. The profit on a bowl is $40 and $50 on a pitcher. She works 40 hours weekly, has 120 pounds of clay available each week, and wants more profits.
Max Z = 40x + 50y profits
s.t.
1x + 2y  40 hours
4x + 3y  120 clay
x, y  0 non-negativity

15. Diet Example (lbs / bag) Nitrogen Phosphate SuperGro 2 4 CropKwik 3
“Diet” Problem. A farmer is preparing to plant a crop in the spring and needs to fertilize a field. There are two brands of fertilizer he can use: SuperGro and CropKwik. Each brand has a specific amount of nitrogen and phosphate. The field requires at least 16 pounds of nitrogen and 24 pounds of phosphate. SuperGro costs $6 per bag and CropKwik $3. How many bags of each type should the farmer use to adequately fertilize his field?
(lbs / bag)
Nitrogen
Phosphate
SuperGro 2 4
CropKwik 3

16. Investment Example
Portfolio Management. Kathleen Allen has $70,000 to invest. She can invest in municipal bonds (8.5% annual return), CD’s (5%), treasury bills (6.5%), or in a growth stock fund (13%). She has established the following guidelines to manage her risk and diversify her portfolio:
No more than 20% in municipal bonds
CD’s can be no more than sum of the other three
At least 30% must be in CD’s and treasury bills
The sum of treasury bills and CD’s must be at least 120% of the sum invested in bonds and stock
All $70,000 must be invested

17. Cash Flow LP
Cash Flow Problem. Toyz.com is a large online retailer of toys. Projected revenues and payables ($ millions) are shown below for the next 6 months. It can take out a 6 month loan at an annual rate of 10%, or can borrow for a month at a time for 16%. What loan schedule will minimize interest payments?

18. Employee Scheduling
A restaurant must create a wait staff schedule each week. Employees work 6 hours per day (plus 2 for setup and cleanup) 5 consecutive days, then have 2 days off. What schedule will minimize costs?

19. Production/Marketing LP
Western Slope Apples produces apple juice and sauce. Juice costs $0.60 to produce and sells for $1.45 per jar. Sauce costs $0.85 and sells for $1.75 per bottle. Sauce must be at least 30% but not more than 60% of production. “Natural” demand for sauce is 5,000 jars plus 3 jars for each $1 spent on advertising. Natural demand for juice is 4,000 bottles plus 5 bottles per $1 of advertising. WSA has a total budget of $16,000 for production and advertising, and wants to maximize profits.

20. Further Study SYST 6070 – Survey of Operations Research
Linear programming
Integer, nonlinear programming
Decision theory
Queueing analysis
Game theory
Networks
Topics with which every business student should be familiar!

21. Competing with Quality
Next Week:
Competing with Quality