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Linear Programming

Linear Programming. Operations Management Dr. Ron Lembke. Motivating Example.

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Linear Programming

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  1. Linear Programming Operations Management Dr. Ron Lembke

  2. Motivating Example • Suppose you are an entrepreneur making plans to make a killing over the summer by traveling across the country selling products you design and manufacture yourself. To be more straightforward, you plan to follow the Dead all summer, selling tie-dyed t-shirts and screenprinted sweatshirts.

  3. Example • You are really good with tie-dye, so you earn a profit of $25 for each t-shirt. • The sweatshirt screen-printed sweatshirt makes a profit of $20. • You have 4 days before you leave, and you want to figure out how many of each to make before you head out for the summer. • You plan to work 14 hours a day on this. It takes you 30 minutes per tie dye, and 15 minutes to make a sweatshirt.

  4. Example You have a limited amount of space in the van. Being an engineer at heart, you figure: • If you cram everything in the van, you have 40 cubic feet of space in the van. • A tightly packed t-shirt takes 0.2 ft3 • A tightly packed sweatshirt takes 0.5 ft3. How many of each should you make?

  5. Summary 14 hrs / day Van: 40.0 ft3 4 days Tshirt: 0.2 ft3 30 min / tshirt Sshirt: 0.5 ft3 15 min / Sshirt How many should we make of each?

  6. Trial and Error • Use up all of the space? • Sweatshirts: 40/0.5 = 80. 80*20 = $1,600 • T-shirts: 40/0.2 = 200! 200*25 = $5,000 cool! • Use all of your time? • Ss: 56/0.25 = 224. 224 * $20 = $4,480 • Ts: 56/0.5 = 112. $25*112 = $2,800 • Fill it with Tshirts? Only time to make 112 • Spend all your time making Ss? Only space for 80

  7. Trial and Error S T Space Time Profits Comments 8 0 0 40 20 $1,600 << 56hr 0 200 40 100 $5,000 > 56 hrs 224 0 112 56 $4,480 > 40 cu ft 0 112 22.4 56 $2,800 << 40 cu ft

  8. Improving the Solution • (0,112) all time is used, van not full $2,800 • Look for a compromise solution • What if make one less T? • Frees up 0.5 hrs, revenue goes down $25 • In 0.5hrs, could make 2 S, brings in $40 more • Same amt of time, $15 more! • 1 T less frees up 0.2 ft3 2 S add 1.0 ft3 • Increase 0.8 ft3 van wasn’t full, so no problem • Trade 1 T for 2 S, gain $15! $2,815

  9. Improving Solution • Keep making trade. How many times? • Use up 0.8 more space • At (0,112) using 22.4, so 40 – 22.4 = 17.6 avail • 17.6/0.8 = 22 Make trade 22 times • (0,112) + 44S – 22T = (44,90) • Space 44*0.5 + 90*0.2 = 22+18 = 40 cu ft • Time 44*0.25 + 90*0.5 = 11 + 45 = 56 hrs • Van is full, all the time is used • Profits 44*20 + 90*25 = 880 + 2250=$3,130

  10. Write down the problem • We could express the problem like this: 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

  11. Linear Programming • What we have just done is called “Linear Programming.” • Has nothing to do with computer programming • Invented in WWII to optimize military “programs.” • “Linear” because no x3, cosines, x*y, etc.

  12. Standard Form • Linear programs are written the following way: Max 3x + 4y s.t. x + y <= 10 x + 2y <= 12 x >= 0 y >= 0

  13. Standard Form Objective Coefficients • Linear programs are written the following way: Objective Function Max 3x + 4y s.t. x + y <= 10 x + 2y <= 12 x >= 0 y >= 0 Constraints RHS (right hand side) Non-negativity Constraints LHS (left hand side) inequalities

  14. Summary Solved a linear program Wrote the problem mathematically, in “standard form” Solved the problem using trial and error

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