Introduction to Linear Programming Professor Stephen Lawrence Leeds School of Business University of Colorado Boulder, CO 80309-0419

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

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.

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

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

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…

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

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

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

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

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

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

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

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

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

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?

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?

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.

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!

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