A jumbo-cargo plane has three compartments for storing cargo: front, center and back. These have limited weight (in tons) and space (in cubic meters) capacities as shown in table-1. Further, the weight of the cargo in the respective compartments must be the same proportion of that compartment's weight capacity to maintain the balance of the airplane.

Three cargoes have been offered for shipment on an upcoming flight as space is available. The details are given in Table-2.

Any portion of these cargo can be accepted. The objective is to determine how much of each cargo should be accepted and how to distribute each among the compartments to maximize the total profit for the flight.

You are expected to use R and lpSolveAPI (or any other solver of your choice) to solve this problem. For variables, we suggest the following: Let Xi,j be the amount of cargo i in the compartment j, where i{1,2,3}and j{Front,Center,Back}.The optimum solution is: Let X i , j be the amount of cargo i in the compartment j , where i { 1 , 2 , 3 } and j { F r o n t , C e n t e r , B a c k } . The optimum solution is:

Question 3 options:

: X1,Front=0tonsX1,Center=0tonsX1,Back=0tonsX2,Front=14.81tonsX2,Center=1.11tonsX2,Back=14tonsX3,Front=0.74tonsX3,Center=10.55tonsX3,Back=0tonsTotalprofit=$18,748

A) X 1 , F r o n t = 0 t o n s X 1 , C e n t e r = 0 t o n s X 1 , B a c k = 0 t o n s X 2 , F r o n t = 14.81 t o n s X 2 , C e n t e r = 1.11 t o n s X 2 , B a c k = 14 t o n s X 3 , F r o n t = 0.74 t o B) n s X 3 , Center=10.55tons X 3 , B a c k = 0 t o n s T o t a l p r o f i t = $ 18 , 748 X1,Front=0tons

C) X1,Center=0tonsX1,Back=0tonsX2,Front=16tonsX2,Center=2tonsX2,Back=0tonsX3,Front=0tonsX3,Center=10tonsX3,Back=8.75tonsTotalprofit=$18,450

D) X1,Front=5.04tonsX1,Center=0tonsX1,Back=3.53tonsX2,Front=8.95tons X2,Center=0tonsX2,Back=9.05tonsX3,Front=0tons X3,Center=10.49tons X3,Back=0tonsTotalprofit=$17,779