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SOLVING LINEAR PROGRAMS

Dr. Ron Lembke. SOLVING LINEAR PROGRAMS. Solving LPs. Need a more formal method How to find points to solve for? A picture is worth a thousand words. Standard Form. Linear programs are written the following way: . Max 20 S + 25 T s.t . 0.5 S + 0.2 T <= 40 0.25 S + 0.5 T <= 56

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SOLVING LINEAR PROGRAMS

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  1. Dr. Ron Lembke SOLVING LINEAR PROGRAMS

  2. Solving LPs Need a more formal method How to find points to solve for? A picture is worth a thousand words

  3. Standard Form • Linear programs are written the following way: Max 20 S + 25 T s.t. 0.5 S +0.2 T <= 40 0.25 S + 0.5 T <= 56 S >= 0 T >= 0 Space Time

  4. Graphical Solution 100 Draw a picture to represent the feasible combinations of S and T, that meet all of the constraints, and find the best point Draw a line of all points that would exactly use up all of the space, and a line of all the points that would exactly use up all of the time 80 60 T T-shirts 40 20 0 0 20 40 60 80 S Sweatshirts

  5. Endpoints of constraints Space: 0.5 S + 0.2 T <= 40 If S=0, then 0.2T=40, so T=200 (0,200) If T=0, then 0.5S=40, so S=80 (80,0) Time: 0.25 S + 0.5 T <= 56 If S=0, 0.5T=56, so T=112 (0,112) If T=0, 0.25S=56, so S=224 (224,0) Hey! These look familiar!

  6. Graphical Solution S=0, T=200 Draw a line to represent Space constraint 200 160 120 T T-shirts Space Constraint 80 S=80, T=0 40 0 0 50100150200 S Sweatshirts

  7. Graphical Solution Draw a line to represent Time constraint 200 160 S=0, T=112 120 T T-shirts 80 S=224, T=0 Time Constraint 40 0 0 50100150200 S Sweatshirts

  8. Graphical Solution Green area satisfies all constraints “Feasible Region” 200 160 S=0, T=112 120 T Space T-shirts 80 Time Constraint S=80, T=0 40 0 0 50100150200 S Sweatshirts

  9. Graphical Solution Last time, we started at (0,112) in search of a better solution. We traded 1 T for 2S, which kept the amount of time used constant, and kept doing that until we used up all of the space (that is, hit the space constraint). 200 160 S=0, T=112 120 T Space T-shirts 80 Time Constraint S=80, T=0 40 0 0 50100150200 S Sweatshirts

  10. Graphical Solution Where do constraints intersect? Where the lines cross - the point is on both 0.5 S + 0.2 T= 40 0.25S + 0.5T= 56 So for S and T. Two equations, two unknowns Algebra! Woohoo!! 200 160 S=0, T=112 120 T Space T-shirts 80 Time Constraint S=80, T=0 40 0 0 50100150200 S Sweatshirts

  11. Graphical Solution Multiply one (or both) to get same coefficients 0.5 S + 0.2 T= 40 [0.25S+ 0.5T = 56] x2 becomes 0.5S + 1 T = 112 -[0.5S + 0.2T = 40] subtract the first one ----------------------------- 0S + 0.8T = 72, so T = 72/0.8 = 90 Substitute into any of the three equations: 0.5S + 0.2*90 = 40, so 0.5S = 40-18 = 22 So S =22/0.5 = 44 Optimal = (44, 90) 200 160 S=0, T=112 120 T Space T-shirts Time 80 S=80, T=0 40 0 0 50100150200 S Sweatshirts

  12. Evaluate Profits at Intersection Points 200 S=0, T=112 Z=20*0+25*112 = 2,800 160 S = 44, T=90 Z=20*44 + 25*90 = 880 + 2250 = 3,130 120 T Space T-shirts Time 80 S=80, T=0 Z=20*80 +25*0 = 1600 40 0 0 50100150200 S Sweatshirts

  13. Edge Points Better than Interior Points 200 A has to be better than B C has to be better than B To figure out if A or C is better, have to evaluate the objective function at both. 160 120 A 80 B C 40 0 0 50100150200

  14. Corner Points Better than Middle Edge 200 IF E is better than D, then F will be better than D or E Or, IF E is better than F, D will be better than E or F. Using same logic, either F or H has to be better than G. One has to be better than G. 160 120 D E F 80 G 40 H 0 0 50100150200

  15. Find all the Corner Points of Feasible Space 200 There can be many, many constraints. Find the feasible space. Ignore constraints that don’t border it. Constraints can be vertical or horizontal. 160 120 80 40 0 0 50100150200

  16. Solution Method (For 2 variables only) Find end points of each constraint Draw constraints Figure out which intersections are interesting Use algebra to solve for intersection pts Find profits (or costs) of intersections Choose the best one

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