Big M Simplex Method Travel Agent

B

BeanbaG

New member
Joined
Mar 6, 2013
Messages
2
Good day everyone,


I have been struggling with this question for weeks, but cannot find a solution.
I cannot formulate the maximisation equation properly and/or the constraints



. Here is the question:

A travel agent is planning a charter trip to a popular sea resort. The 10-day, 9-night package includes the fare for the round-trip travel, surface transportation, board and lodging and selected tour options. The charter trip is restricted to 300 persons and past experience indicates that there will be no problem in getting 300 people. The problem for the travel agent is to determine the number of Deluxe, Standard and Economy packages to offer for this charter. These three plans each differ according to the seating and service on the flight, quality of accommodation, meal plans and tour options. The following table summarizes the estimated price for the three packages and the corresponding expenses for the travel agent per person. The travel agent has hired an aircraft for a flat fee of $250000 for the entire trip.

Tour Plan
Price
($)
Hotel Costs
($)
Meals and other Expenses ($)
Deluxe
10000
3500
4500
Standard
7500
2500
3000
Economy
6500
2000
2500




In planning the trip the following considerations must be taken into account:
1. At least 10% of the packages must be of the deluxe type.
　
2. At least 35% but not more than 70% must be of the standard type
　
3. At last 30% must be of the Economy type.
　
4. The maximum number of deluxe packages available in any aircraft is restricted to 100.
　
5. The hotel desires that at least 150 tourists should be on the deluxe and Standard packages together.
[FONT=Calibri,Calibri][FONT=Calibri,Calibri]Required:
[/FONT][/FONT]Use the simplex method to determine the number of packages to offer in each type so as to maximize profits.

==========================================================================
Let No of Deluxe = x1, Standard = x2, Economy = x3. Therefore Deluxe: 10000 - (3500 + 4500) = 2000 etc.
So Maximise: 2000x1 +2000x2 + 2000x3 - 250 000 subject to 30<= x1 <= 100 ; 105 <= x2 <= 210 ; x3 >= 90 ; x1 +x2 >= 150 ; x1 +x2 +x3 <= 300 where x1, x2, x3 > 0. I also introduced slack and artificial variables as part of my solution. See attached pdf. I would like someone to confirm my contraints and maximisation equation as a start. I will then use the bigM substitution method to obtain a solution. I would really appreciate help in this, im quite desperate and have been stuck for so long :/
 

Attachments

  • Book2 Constraints.pdf
    14.7 KB · Views: 12
BeanbaG said:
Good day everyone,


I have been struggling with this question for weeks, but cannot find a solution.
I cannot formulate the maximisation equation properly and/or the constraints



. Here is the question:

A travel agent is planning a charter trip to a popular sea resort. The 10-day, 9-night package includes the fare for the round-trip travel, surface transportation, board and lodging and selected tour options. The charter trip is restricted to 300 persons and past experience indicates that there will be no problem in getting 300 people. The problem for the travel agent is to determine the number of Deluxe, Standard and Economy packages to offer for this charter. These three plans each differ according to the seating and service on the flight, quality of accommodation, meal plans and tour options. The following table summarizes the estimated price for the three packages and the corresponding expenses for the travel agent per person. The travel agent has hired an aircraft for a flat fee of $250000 for the entire trip.

Tour Plan Price
($) 		Hotel Costs
($) 		Meals and other Expenses ($)
Deluxe 10000 3500 4500
Standard 7500 2500 3000
Economy 6500 2000 2500




In planning the trip the following considerations must be taken into account:
1. At least 10% of the packages must be of the deluxe type.
　
2. At least 35% but not more than 70% must be of the standard type
　
3. At last 30% must be of the Economy type.
　
4. The maximum number of deluxe packages available in any aircraft is restricted to 100.
　
5. The hotel desires that at least 150 tourists should be on the deluxe and Standard packages together.
Required:
Use the simplex method to determine the number of packages to offer in each type so as to maximize profits.

==========================================================================
Let No of Deluxe = x1, Standard = x2, Economy = x3. Therefore Deluxe: 10000 - (3500 + 4500) = 2000 etc.
So Maximise: 2000x1 +2000x2 + 2000x3 - 250 000 subject to 30<= x1 <= 100 ; 105 <= x2 <= 210 ; x3 >= 90 ; x1 +x2 >= 150 ; x1 +x2 +x3 <= 300 where x1, x2, x3 > 0. I also introduced slack and artificial variables as part of my solution. See attached pdf. I would like someone to confirm my contraints and maximisation equation as a start. I will then use the bigM substitution method to obtain a solution. I would really appreciate help in this, im quite desperate and have been stuck for so long :/
Click to expand...
Since the profit is identical for all three classes, the maximization becomes (x1+x2+x3)=300. Note the statement, there will be no problem in getting 300 people. That looks like a constraint! In fact, you used it like a constraint when you calculated percentages.
1) 30 <= x1
2) 105 <= x2 <= 210
3) 90 <= x3
4) x1 <= 100
5) 150 <= x1 + x2

Suppose we eliminate x3 as a variable by setting x3 = 300 - x1 - x2. Then constraint 3) becomes
3') 210 <= x1 + x2
But this overrides 5), so we have
1,4) 30 <= x1 <= 100, 0 <= x1' <= 70
2) 105 <= x2 <= 210, 0 <= x2' <= 105
3') 210 <= (x1 + x2), 75 <= (x1' + x2')

Plotting 3') in either the primed or the unprimed coordinates, I find a trapezoidal region that satisfies all constraints. Because the maximization function is independent of the variables, there is not a unique solution to the problem as stated.
 
DrPhil said:
Since the profit is identical for all three classes, the maximization becomes (x1+x2+x3)=300. Note the statement, there will be no problem in getting 300 people. That looks like a constraint! In fact, you used it like a constraint when you calculated percentages.
1) 30 <= x1
2) 105 <= x2 <= 210
3) 90 <= x3
4) x1 <= 100
5) 150 <= x1 + x2

Suppose we eliminate x3 as a variable by setting x3 = 300 - x1 - x2. Then constraint 3) becomes
3') 210 <= x1 + x2
But this overrides 5), so we have
1,4) 30 <= x1 <= 100, 0 <= x1' <= 70
2) 105 <= x2 <= 210, 0 <= x2' <= 105
3') 210 <= (x1 + x2), 75 <= (x1' + x2')

Plotting 3') in either the primed or the unprimed coordinates, I find a trapezoidal region that satisfies all constraints. Because the maximization function is independent of the variables, there is not a unique solution to the problem as stated.
Click to expand...

Hi DrPhil, thanks for the reply. I agree with you , 100%. However, I was specifically told that "the x1+x2+x3 = 300 constraint is wrong"
And thus have no idea how to approach the problem.
Should it be done on a percentage basis, for example, 1) becomes
0.1(x1+x2+x3)<=x1
-0.9*x1+0.1*x2+0.1*x3<=0
 
I think the constraint should be: 0 ≤ x1 + x2 + x3 ≤ 300
 
You must log in or register to reply here.
Top