Introduction to Linear Programming

1
Introduction to Linear Programming

• Professor Stephen Lawrence
• Leeds School of Business
• University of Colorado
• Boulder, CO 80309-0419

2
Agenda

• Examples of LPs
• Graphical solution method
• Standard form
• Assumptions of LP
• Solving LPs 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 4 3
5
LP Marketing Example
Advertising Mix. Follys 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
9
Assumptions of LP

• Linear objective function, constraints
• 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 LPs 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
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
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 4 3
16
Investment Example
Portfolio Mix. Kathleen Allen has 70,000 to
invest. She can invest in municipal bonds (8.5
annual return), CDs (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
• CDs can be no more than sum of the other three
• At least 30 must be in CDs and treasury bills
• The sum of treasury bills and CDs 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