7-14 The Electrocomp Corporation manufactures two electrical products: air conditioners and large fans. The assembly process for each is similar in that both require a certain amount of wiring and drilling. Each air conditioner takes 3 hours of wiring and 2 hours of drilling. Each fan must go through 2 hours of wiring and 1 hour of drilling. During the next production period, 240 hours of wiring time are available and up to 140 hours of drilling time may be used. Each air conditioner sold yields a profit of $25. Each fan assembled may be sold for a $15 profit. Formulate and solve this LP production mix situation to find the best combination of air conditioners and fans that yields the highest profit. Use the corner point graphical approach. X1 = the number of air conditioners scheduled to be produced X2 = the number of fans scheduled to be produced Maximize 25X1 + 15X2 (maximize profit) subject to: 3X1 + 2X2 240 2X1 +X2 140 (wiring capacity constraint) (drilling capacity constraint) X1 , X2 0 (non-negativity constraints) Optimal Solution: X 1 = 40 X2 = 60 Profit = $1,900 7-15 Electrocomp's management realizes that it forgot to include two critical constraints (see Problem7-14). In particular, management decides that there should be a minimum number of air conditioners produced in order to fulfill a contract. Also, due to an over supply of fans in the proceeding period, a limit should be placed on the total number of fans produced. *** If Electrocomp decides that at least 20 air conditioners should be produced, but no more than 80 fans should be produced, what would be the optimal solution? How much slack or surplus is there for each of the 4 constraints? *** If Electrocomp decides that at least 30 air conditioners should be produced but no more than 50 fans should be produced, what would be the optimal solution? How much slack or surplus is there for each of the 4 constraints of the optimal solution? X 1 = the number of air conditioners scheduled to be produced X2 = the number of fans scheduled to be produced Maximize 25x1 + 15x2 (maximize profit) Subject to 3x1 + 2x2 240 (wiring capacity constraint) 2X1 +X2 140 (drilling capacity constraint X1 20 (a/c contract constraint) X2 80 (maximum # of fans constraint) X1 , X2 0 (non-negativity constraints) Optimal Solution: X 1 = 40 X2 = 60 Profit = $1,900 X1 = the number of air conditioners scheduled to be produced X2 = the number of fans scheduled to be produced Maximize 25x1 + 15x2 (maximize profit) Subject to 3x1 + 2x2 235 (wiring capacity constraint) 2X1 +X2 140 (drilling capacity constraint X1 45 (a/c contract constraint) X2 50 (maximum # of fans constraint) Profit 25 (45) + 15 (50) = $ 1875 Constraint Wiring hours Drilling hours Fan Production A/C Slack 0 0 0 Surplus 15 7-18 The Dean of the Western College of Business must plan to school's course offerings for the fall semester. Student demands make it necessary to offer at least 30 undergraduate and 20 graduate courses in the term. Faculty contracts also dictate that at least 60 courses be offered in total. Each undergraduate course taught costs the college an average of $2,500 in faculty wages, and each graduate course costs $3,000. How many undergraduate and graduate courses should be taught in the fall so that total faculty salaries are kept to a minimum? X1 = the number of undergraduate courses scheduled X2= the number of graduate courses scheduled Minimize 2,500X1 + 3,000X2 (minimize faculty salaries) Subject to: X1 > 30 (schedule at least 30 undergraduate courses) X2 > 20 (schedule at least 20 grad courses) X1 + X2 > 60 (schedule at least 60 total courses) X1, X2 > 0 (non negativity constraints) Optimal Solution: X1 = 40 X2 = 20 Cost $160,000