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LP: Linear Programming Mathematical Model

LP: Linear Programming Mathematical Model. Optimize a linear function of many variables, under linear inequalities. The function to optimize is called the economic function The linear inequalities are called constraints The variables are called decision variables

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LP: Linear Programming Mathematical Model

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  1. LP: Linear Programming Mathematical Model • Optimize a linear function of many variables, under linear inequalities. • The function to optimize is called the economic function • The linear inequalities are called constraints • The variables are called decision variables • Optimizing means finding for the decision variables the values that maximize or minimize (according to context) the economic function Max Z = c1x1 + c2x2 + … + cnxn subject to a11x1 + a12x2 + … + a1nxnb1 a21x1 + a22x2 + … + a2nxnb2 … am1x1 + am2x2 + … + amnxnbm x1 0, x2 0, …, xn 0

  2. Example of LP: Product Mix 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

  3. Graphical Solution 40 (40, 50) 20 24, 8 30 40

  4. Example of LP: Diet 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 model of the Diet example x1 = bags of SuperGro x2 = bags of CropKwik Min Z = 6x1 + 3x2 s.t. 2x1 + 4x2 16 4x1 + 3x2  24 x1 , x2  0

  6. Graphical Solution 8 (-6, -3) 4 4.8 ,1.6 6 8

  7. LP Marketing Example 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

  8. Example of LP: Investment (portfolio diversification) 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

  9. LP Model of the Investment Example x1 = bonds x3 = T-Bills x2 = CD’s x4=stocks Max Z = 0.085x1 + 0.05x2 + 0.065x3 + 0.13x4 returns s.t. x1 0.2 (x1 + x2 + x3 + x4) bonds x2x1 + x3 + x4 CD’s x2 + x3 0.3 (x1 + x2 + x3 + x4) CD’s, T-Bills x2 + x3 1.2(x1 + x4) CD’s, T-Bills x1 + x2 + x3 + x4 = 70,000 investment x1 , x2 , x3 , x4 0

  10. 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…

  11. Assumptions of LP • Linear objective function, linear constraints • Proportionality • Additivity • Divisibility • Continuous decision variables • If not, we fall in a variant called “integer programming” • Certainty • Deterministic parameters

  12. LP Concepts • Decision variables • Objective function • Constraints • Feasible solutions • Feasible region: a convex polytope (polygone in 2D, polyhedron in 3D etc..) • Corner point solutions • Optimal solution • “Constrained optimization”

  13. Cash Flow LP 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?

  14. 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?

  15. 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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