Introduction to Linear Programming

Professor Stephen LawrenceLeeds School of BusinessUniversity of ColoradoBoulder, CO 80309-0419

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

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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.

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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

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LP Marketing Example

$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

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:

Exposure Cost

TV 20,000 $15,000

Radio 12,000 6,000

Print 9,000 4,000

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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…

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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

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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

j =1

n

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Assumptions of LP Linear objective function,

constraints Proportionality Additivity

Divisibility Continuous decision variables

Certainty Deterministic parameters

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

Solving LP’s with Excel Solver

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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

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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

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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 profitss.t.

1x + 2y 40 hours4x + 3y 120 clayx, y 0 non-negativity

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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

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Investment Example

• 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

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:

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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?

Month Revenues PayablesJuly 20 60 August 30 60 September 40 80 October 50 30 November 80 30 December 100 20

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Employee SchedulingA 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?

DayRqdHours Meetings

MeetingHours

TotalHours

Mon 125 1 24 149Tue 200 0 0 200Wed 350 1 24 374Thu 300 0 0 300Fri 650 3 52 702Sat 725 4 64 789Sun 250 2 36 286

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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.

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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!

Competing with Quality

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