Exercise 2: Airlines Corporation (Pg. 525)
An airlines corporation is considering the purchase of jet passenger planes so as to increase their passenger service. The type A plane costs $450 million each, the type B costs $400 million each, and the type C costs $250 million each. The corporation has budgeted $50 billion for the purchase of these planes in the forthcoming financial year. The three types of planes, if purchased, would be utilized at essentially maximum capacity. It is estimated that the net annual profit resulting from utilization of these planes would be $15 million for type A, $10.5 million for type B, and $7.5 million for type C. It is estimated that 25 trained pilots will be available, and if only C type planes were purchased, the maintenance facilities would be able to handle 30 new planes. However, each type B plane is equivalent to 4/3 type C plane and each type A plane is equivalent to 5/3 type C planes in terms of their use of maintenance facilities.
Develop a linear optimization model to determine how many of each type of plane should be purchased.
Implement your model on a spreadsheet and find an optimal solution.
Step 1: Formulate the Linear programming Model
xA: type A plan
xB: type B plan
xC: type C plan
Maximize profit:
P = 15000000xA + 10500000xB + 7500000*xC
SUBJECT TO:
budget next year: $50B
50000000xA + 400000000xB + 250000000*xC <= 50000000000
pilots: 25
xA + xB + xC <= 25
maintenance facilities:
xC <= 30
xB - 4/3xC = 0
xA - 5/3xC = 0
Step 2: Using lpSolve package of R to solve
library(lpSolve)
obj.fun <- c(15000000, 10500000, 7500000)
constr <- matrix(c(50000000, 400000000, 250000000, 1, 1, 1, 0, 0, 1, 0, 1, -4/3, 1, 0, -5/3), ncol = 3, byrow= TRUE)
constr.dir <- c("<=", "<=", "<=", "=", "=")
rhs <- c(50000000000, 25, 30, 0, 0)
prod.sol <- lp("max", obj.fun, constr, constr.dir, rhs, compute.sens = TRUE, all.int=TRUE)
prod.sol## Success: the objective function is 2.79e+08prod.sol$solution #decision variables values## [1] 10 8 6Conclusion: to maximize profit, we should buy 10 plan A, 8 plan B and 6 plan C. Maximum profit is $279 million.