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Intro Management Science

Intro Management Science. 472.21 2 Fall 2011 Bruce Duggan Providence University College. This Week. Review Cases from ch 1 Linear Programming ch 2 formulas & graphs. Case 1: Clean Clothes Corner. Current volume? she’s just breaking even. Case 1: Clean Clothes Corner.

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Intro Management Science

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  1. Intro Management Science 472.21 2 Fall 2011 Bruce Duggan Providence University College

  2. This Week • Review • Cases from ch 1 • Linear Programming • ch 2 • formulas & graphs

  3. Case 1: Clean Clothes Corner • Current volume? • she’s just breaking even

  4. Case 1: Clean Clothes Corner • B. Increase needed to break even?

  5. Case 1: Clean Clothes Corner • C. Monthly profit? Z = vp - cf - vcv Z = 4,300.00  $1.10 - ($1,700.00 + $1,350.00) - 4,300  $0.25

  6. Case 1: Clean Clothes Corner • D. If lower price? BE? Z? Z = vp - cf - vcv

  7. Case 1: Clean Clothes Corner • E. Which is the better choice? Z with new equipment? Z without new equipment?

  8. Case 2: Ocobee • Which option is better? • make the rafts yourself? • buy them from North Carolina?

  9. ch 2: Linear Programming George Dantzig http://forum.stanford.edu/blog/?p=27

  10. Linear Programming • Jargon • Linear programming • l.p. • “figuring stuff out with basic algebra” • Model formulation • Stating our problem in words/math/graphs • Sensitivity analysis • “What happens if…?”

  11. Linear Programming • Jargon • Why is there jargon? handout

  12. Applications • Kellogg • pg 35 • Nutrition Coordinating Center • pg 46 • Soquimich • pg 51

  13. Example: Maximization • The St. Adolphe Historical Museum • We have a group of older volunteers • The St. Adolphe Craft League • They’ve offered to make toothpick tchochkes to sell at the gift shop • Red River ox carts • the first church in St. Adolphe • We can sell everything they make

  14. St. Adolphe Craft League • They want to know: • How many ox carts? • How many churches? • Goal • To make the most profit possible for the museum

  15. St. Adolphe Craft League • Resource availability • 40 hrs of labor • 120 boxes of toothpicks • Decision variables • x1 = number of ox carts to make • x2 = number of churches to make

  16. Resource Requirements • Product • Profit ($/unit) • cart • 40 • church • 50 St. Adolphe Craft League • Product resource requirements and unit profit: • Labour (hr/unit) • Material (boxes/unit) • 1 • 4 • 2 • 3

  17. St. Adolphe Craft League • Objective function • Maximize Z = $40x1 + $50x2 • Resource constraints • 1x1 + 2x2 40 hours of labor • 4x1 + 3x2 120 boxes of toothpicks • Non-Negativity constraints • x1 0; x2 0

  18. St. Adolphe Craft League • Problem definition • Complete linear programming model Maximize Z = $40x1 + $50x2 subject to: 1x1 + 2x2 40 4x1 + 3x2 120 x1, x2 0

  19. St. Adolphe Craft League • Model formulation: • l.p. Max Z = $40x1 + $50x2 s.t. 1x1 + 2x2 40 4x1 + 3x2 120 x1, x2 0 no computers yet

  20. St. Adolphe Craft League Max Z = $40x1 + $50x2 s.t. 1x1 + 2x2 40 4x1 + 3x2 120 x1, x2 0 words math graphs

  21. x2 40 30 20 10 x1 10 20 30 40 0 St. Adolphe Craft League Max Z = $40x1 + $50x2 s.t. 1x1 + 2x2 40 4x1 + 3x2 120 x1, x2 0

  22. x2 40 30 20 10 x1 10 20 30 40 0 St. Adolphe Craft League Max Z = $40x1 + $50x2 s.t. 1x1 + 2x2 40 4x1 + 3x2 120 x1, x2 0

  23. x2 40 30 20 10 x1 10 20 30 40 0 St. Adolphe Craft League Max Z = $40x1 + $50x2 s.t. 1x1 + 2x2 40 4x1 + 3x2 120 x1, x2 0

  24. x2 40 30 20 10 x1 10 20 30 40 0 St. Adolphe Craft League Max Z = $40x1 + $50x2 s.t. 1x1 + 2x2 40 4x1 + 3x2 120 x1, x2 0

  25. x2 40 x1 = 0 ox carts x2 = 20 churches Z = $1,000 x1 = 24 ox carts x2 = 8 churches Z = $1,360 30 20 10 x1 = 30 ox carts x2 = 0 churches Z = $1,200 x1 10 20 30 40 0 St. Adolphe Craft League Max Z = $40x1 + $50x2 s.t. 1x1 + 2x2 40 4x1 + 3x2 120 x1, x2 0

  26. Linear Programming • lp has 2 main tools • maximization • most profit • minimization • least cost Z means profit Z means cost

  27. words graphs math Example: Minimization • Friesen Farms • section of land needs at least • 16 lb nitrogen • 24 lb phosphate • 2 brands of fertilizer available • DeSallaberry Superior • Carmen Crop • Goal • Meet fertilizer needs at minimum cost • Problem • How much of each brand should you buy?

  28. words graphs math Friesen Farms

  29. words graphs math Friesen Farms

  30. words graphs math Friesen Farms • Objective function • Minimize Z = $6x1 + $3x2 • Decision variables • x1 = bags of DeSallaberry to buy • x2 = bags of Carmen to buy

  31. words graphs math Friesen Farms • Objective function • Minimize Z = $6x1 + $3x2 • Model constraints • 2x1 + 4x2 16 (lb) nitrogen constraint • 4x1 + 3x2 24 (lb) phosphate constraint • x1,x2 0 non-negativity constraint

  32. words graphs math Friesen Farms • Model formulation: • l.p. Min Z = $6x1 + $3x2 s.t. 2x1 + 4x2≥ 16 4x1 + 3x2≥ 24 x1, x2 0

  33. words graphs math x2 8 6 4 2 x1 2 4 6 8 0 Friesen Farms Min Z = $6x1 + 3x2 s.t.2x1 + 4x2≥ 16 4x1 + 3x2≥ 24 x1, x2 0

  34. x2 8 6 4 2 x1 2 4 6 8 0 Friesen Farms Min Z = $6x1 + 3x2 s.t.2x1 + 4x2≥ 16 4x1 + 3x2≥ 24 x1, x2 0

  35. x2 8 6 4 2 x1 2 4 6 8 0 Friesen Farms Min Z = $6x1 + 3x2 s.t.2x1 + 4x2≥ 16 4x1 + 3x2≥ 24 x1, x2 0

  36. x1 = 0 bags of DeSallaberry x2 = 8 bags of Carmen Z = $24 x2 8 x1 = 5 DeSallaberry x2 = 2 Carmen Z = $36 6 4 x1 = 8 DeSallaberry x2 = 0 Carmen Z = $48 2 x1 2 4 6 8 0 Friesen Farms Min Z = $6x1 + 3x2 s.t.2x1 + 4x2≥ 16 4x1 + 3x2≥ 24 x1, x2 0

  37. x2 8 6 4 2 x1 2 4 6 8 0 Friesen Farms Min Z = $6x1 + 3x2 s.t.2x1 + 4x2≥ 16 4x1 + 3x2≥ 24 x1, x2 0

  38. x2 8 6 4 2 x1 2 4 6 8 0 Friesen Farms Min Z = $6x1 + 3x2 s.t.2x1 + 4x2≥ 16 4x1 + 3x2≥ 24 x1, x2 0

  39. x1 = 0 bags of DeSallaberry x2 = 8 bags of Carmen s1 = 16 lb of nitrogen s2 = 0 lb of phosphate Z = $2400 x2 8 x1 = 4.8 DeSallaberry x2 = 1.6 Carmen s1 = 0 nitrogen s2 = 0 phosphate Z = $3360 6 4 x1 = 8 DeSallaberry x2 = 0 Carmen s1 = 0 nitrogen s2 = 8 phosphate Z = $4800 2 x1 2 4 6 8 0 Friesen Farms Min Z = $6x1 + 3x2 s.t.2x1 + 4x2≥ 16 4x1 + 3x2≥ 24 x1, x2 0 • Surplus Variables • what’s left over • - don’t contribute to • - “slack”

  40. On computer • much easier to do • goals up to now • the idea • the formulas

  41. usual characteristics & limitations clear goal choice amongst alternatives “certainty” non-probabilistic constraints exist relationships linear slope constant additivity divisibility for graphical solution 2 variables l.p.

  42. Assignment • ch 2 problems • in group • 2 • 38 • yourself • 1 • 16

  43. St. Adolphe Craft League • Model formulation: • l.p. Max Z = $40x1 + $50x2 s.t. 1x1 + 2x2 40 4x1 + 3x2 120 x1, x2 0

  44. Next Week • review ch 2 problems • ch 3 • on the computer • sensitivity analysis

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