Can you generate random linear programming problems?

I looked at Linear Programming, and it are problems like this:

You know that Cabinet X costs 10 cents per unit, requires 6 square feet of floor space, and holds 8 cubic feet of files. Cabinet Y costs 20 dollars per unit, requires 8 square feet of floor space, and holds 12 cubic feet of files.

You have been given 140 dollar for this purchase, though you don't have to spend that much. The office has room for no more than 72 square feet of cabinets.

How many of which model should you buy, in order to maximize storage volume?

But you notate them in special form, so that you can genericly describe many of such problems.

Now suppose I replace all the values in this problem (10, 6, 8, 20, 12, 140, 72) with random other numbers, so I randomly create a completely new problem.

• Do they take longer to solve, than to verify the solution?
• Can I check that they are solveable without solving them?
• Can I check how difficult they will be to solve, without solving them?