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 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?

5. 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:

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

16. 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 Mix. 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:

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!

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