# Math Help - Big M, Simplex Method

1. ## Big M, Simplex Method

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.

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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 :/

I have tried using the Simplex solver : PHPSimplex: Simplex method witht he above contraints and maximisat ion equation ,and am told there is no solution

2. ## Re: Big M, Simplex Method

Since each plan has the same profit of \$2000 there is no combination that will maximise your profit, your profit will always be 300*2000-250000

I can give you 1 more constraint though.
Since E=>90
S+D+E=>S+D+90

You know S+D+E=300 so
300=>S+D+90
210=>S+D

And you now know 210=>S+D=>150

edit. I forgot to mention that my notation was D for the number of people buying the deluxe plan, S for standard and E for economy.