# Can someone help me with this math problem for the simplex method?

I and every classmate in my class is having trouble solving this problem. I just need to know what they constraints are so I can solve the problem. He gave us the answer but I've tried everything and I don't get what he gave so I assume my constraints are wrong. This is the problem below.

Solve the following Linear Programming Problem via the Simplex Method:

A city council is working with a hotel developer to build several hotels along its beach-front. The developer has three prototypes: a convention-style hotel with 500 rooms costing \$100 million, a vacation-style hotel with 200 rooms costing \$20 million, and a small hotel with 50 rooms costing \$4 million. The city council wants a total capacity of at least 3,000 rooms and has three other restrictions:

• at most three convention-style hotels;

• at most twice as many small hotels as vacation-style; and

• at least a fifth as many convention-style hotels as vacation-style and small combined.

How many hotels of each type should the council request in order to minimize cost?

Relevance
• Duke
Lv 7
8 years ago

Let x₁ is the number of type C hotels (conventional), x₂ - number of type V hotels (vacation), x₃ - of type S (small), then we'll have the following Integer Linear Programming Problem:

Minimize { Total_cost = 100x₁ + 20x₂ + 4x₃ }, subject to constraints:

500x₁ + 200x₂ + 50x₃ ≥ 3000 (rooms at least);

x₁ ≤ 3 (at most 3 type C);

-2x₂ + x₃ ≤ 0 (ratio S:V at most 2);

x₁ - ⅕(x₂ + x₃) ≥ 0 (ratio C:(V+S) at least ⅕);

x₁, x₂, x₃ - non-negative integers.

The optimal solution I got in 2 iterations by simplex-method was:

minimum_Total_cost = \$400 million, x = 2, y = 10, z = 0, hence type S hotels should not be built. Here is the optimal table:

. . . . . . . . ||100|20| 4 . | 0 . . . . . | 0 | 0 | 0 . . |

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

x₆ = 20 . . . || 0 | 0 | 2 . .|-0.0066..| 0 | 1 |3.33..|

x₅ = 1 . . . . || 0 | 0 | 0.1 |0.00066.| 1 | 0 |0.66..|

x₂ = 10 . . . || 0 | 1 | 0.5 |0.0033...| 0 | 0 |1.66..|

x₁ = 2 . . . . || 1 | 0 |-0.1 |-0.00066| 0 | 0 |-0.66.|

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Value=400 || 0 | 0 |-4 . | -2/15 . . .| 0 | 0 |-100/3|