Answer to Question #194464 in Operations Research for Mr Okay Okay

Question

Question 1

A train can accommodate at most 80 passengers, economy class (x) and first-

class (y) passengers. To avoid making a loss the train must carry at least ten

economy class passengers and at least twenty first class passengers. Due to the

number of complementary items the first-class passengers receive, the number

of first-class passengers should be at most three times the number of economy

class passengers.

First class tickets cost N$5000 while economy class tickets cost N$3000.

i) State/ describe what the variables x and y represent.

ii) State the objective function.

iii) List the constraints.

iv) Draw and clearly label the constraints on graph paper.

v) Clearly show and label the feasible region.

vi) From the graph deduce the number of economy and first-class tickets

that should be sold to make as much money as possible.

vii) Work out the maximum amount of money from ticket sales that the

train company will make.

Answer 1

(i) X and Y represent the number of passenger's in train.

 Let, X= Economy class

    Y=First class

(ii) Objective function-

Z= 5000x+3000y

(iii)

Total number of passengers in train is 80

  So $\Rightarrow X+Y\le80~~~~~~~-(1)$

  $3x\ge y \\ x\ge10\\y\ge25$

(iv)

The Graph is-

(v) The feasible region is shown in the graph-

(vi) There are total 4 situation in this case when values of X and Y varies and these are 

1. (10,25) Z=5000(10)+3000(25)=50000+75000=125000
2. (8.33,25) Z=5000(8.33)+3000(25)=41600+75000=116600
3. (20,60) Z=5000(20)+3000(60)=100000+180000=280000
4. (10,70) Z=5000(10)+3000(70)=50000+210000=260000

Hence we get The maximum value of z at x=20,y=60

Hence Ticket sold for economy class= 20, No. of tickets for first class=60

(vii) The maximum amount Z at(20,60)=280000