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.

4. 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.

5. 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 ft 3  A tightly packed sweatshirt takes 0.5 ft 3. How many of each should you make?

6. Summary 14hrs / day Van:40.0 ft 3 4days Tshirt:0.2 ft 3 30min / tshirt Sshirt:0.5 ft 3 15min / Sshirt How many should we make of each?

7. 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

8. Trial and Error STSpaceTimeProfitsComments 8004020$1,600<< 56hr 020040100$5,000> 56 hrs 224011256$4,480> 40 cu ft 011222.456$2,800<< 40 cu ft

9. 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 ft 3 2 S add 1.0 ft 3  Increase 0.8 ft 3 van wasn’t full, so no problem  Trade 1 T for 2 S, gain $15! $2,815

10. 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

11. Write down the problem We could express the problem like this: Max20S+25T s.t.0.5S+0.2T<=40 0.25S+0.5T<=56 S>=0 T>=0 Space Time

12. 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 x 3, cosines, x*y, etc.

13. Standard Form Linear programs are written the following way: Max3x+4y s.t.x+y<=10 x+2y<=12 x>=0 y>=0

14. Standard Form Linear programs are written the following way: Max3x+4y s.t.x+y<=10 x+2y<=12 x>=0 y>=0 Objective Function Constraints LHS (left hand side) RHS (right hand side) inequalities Non-negativity Constraints Objective Coefficients

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