Please Help. I have to set up linear inequalities including the constraints. I have started with the inequality but I don't know how to get the constraints

Question

This is my question! I have to set up linear inequalities including the constraints from this problem. I have started with the inequality but I am very confused on how to get the constraints.


Problem: A city council is planning to build several hotels along its beach-front. They will be of two types: a convention-style hotel with 400 rooms costing $75 million to build and a vacation-style hotel with 200 rooms costing $40 million. The city council wants a total capacity of at least 3,000 rooms, wants there to be no more than 15 hotels altogether, with at most 8 convention- style, and with no more that twice as many convention-style hotels as vacation-style. How many hotels of each type should the council request in order to minimize cost?

Answer 1

Use a two-dimensional graph with the x coordinate representing the number of convention-style hotels and the y coordinate representing the number of vacation-style hotels.

 

You have four relations that will bound the area of valid solutions.

 

First, the total number of hotels altogether is less that 15:

 

x + y ≤ 15

or

y ≤ -x + 15

 

Second, the total number of rooms is greater that 3000:

 

400x + 200y ≥ 3000

Subtract 400x from both sides and divide by 200 =

y ≥ 2x + 15

 

Third, no more that 8 Convention-style hotels:

 

x ≤ 8

 

Finally, the number of convention-style hotels must be less than or equal to twice the vacation-style hotels:

At most two convention hotels for every vacation hotel.

x ≤ 2y or,

2y ≥ x

y ≥ x/2 

 

Plot each line and shade the area within bounds of each line.

Calculate the points of the intersections of all the lines within the solution area. Use these points to calculate the costs at each intersection using the formula:

 

75x + 45y

 

The point with the lowest cost is best value.

Answer 2

Alexandra,

 

To avoid confusion you have to be methodical in converting each clause in the word problem into a mathematical relation, and place those in a structure (a table here) that will clarify what needs to be done to obtain the answer. So let's begin by defining a few variables:

 

Let x = the number of convention-style hotels with 400 rooms to be built at $75M each.

Let y = the number of vacation-style hotels with 200 rooms to be built at $40M each.

Let C(x,y) = the total cost of building all the hotels given x & y.

 

Then  C(c,y) = 75x + 40y  million dollars for any pair x,y.

 

We'll now number the constraints to refer to them in our table as follows:

Constraint 1:  x>=0 and y>=0  since you can't build a negative number of hotels.

Constraint 2:  x <= 8 since they don't want more than 8 convention-style hotels built.

Constraint 3:  x + y <= 15 since the total number of hotels can't exceed 15.

Constraint 4:  x <= 2y since you want no more than twice as many convention-style than vacation-style hotels.

Constraint 5:  400x + 200y >=3000 or simplified 2x + y >= 15 since they want at least 3000 total rooms.

 

Those are all the constraints.  Now set up a two-dimensional table to help clarify and figure out the admissible values of x & y pairings that satisfy all those constraints and then can be used to compute corresponding C(x,y)s.  Let's make the x-values the rows and possible y-values the columns in this table.  Then Constraints 1 & 2 say that x can only take on the values 0, 1, ... 8.  So that's all the rows we need.  An the y's satisfy noted & preceding constraints.

 

x      y-values that      y-values that       y-values that       y-values that         Corresponding C(x,y) Build Costs

          satisfy #3          satisfy #4           satisfy #5            satisfy ALL

_      __________       __________       __________        ___________        _______________________________

0      0, 1 ... 15           0, 1 ... 15                15                         15                 C(0,15) = $600M

1      0, 1 ... 14           1, 2 ... 14            13 & 14               13 & 14                C(1,13) = $595M; C(1,14)=$635M

2      0, 1 ... 13           1, 2 ... 13            11, 12 & 13         11, 12 & 13

3

4

5

6

7

8 

 

You fill out the rest of this table and then look down the Cost column for the lowest build cost. The corresponding x, y pair then tells you how many hotels of each kind to build.

 

BTW, I didn't say the solution would be quick or not require a lot of manual computation.  That's why linear programming is usually done by computer software, not by hand.  But the software basically mimics an approach like this.  Good luck

Answer 3

Remember, constraints are simply lines.  Let x be the number of Convention-style hotels, and y the number of vacation-style hotels.  Then the inequalities you set up are

1) 400 x + 200 y >= 3000
2) x + y <= 15
3) x <= 8
4) 2y >= x

 

So, we want inequalities that we can graph in the x-y plane.  We can put all the inequalities in slope-intercept form, except for inequality 3, which has infinite slope:

1) 200 y >= -400 x + 3000

           y >= -2 x + 15

2) y <= -x + 15

3) x <= 8

4) y >= x/2

 

So you have a quadrangle here.  Graph the lines, find the four corners of the quandrangle, and one of them is the lowest cost solution.