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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?

Please help. Best Answer gets 10 points. Thanks.

1 Answer

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  • Duke
    Lv 7
    8 years ago
    Favorite Answer

    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|

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