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Re: LinearProgramming[]

  • To: mathgroup at smc.vnet.net
  • Subject: [mg123528] Re: LinearProgramming[]
  • From: Virgil Stokes <vs at it.uu.se>
  • Date: Sat, 10 Dec 2011 07:28:53 -0500 (EST)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com
  • References: <201112091058.FAA03920@smc.vnet.net>

On 09-Dec-2011 11:58, é?? å?? wrote:
> We know all constraints in LinearProgramming[] involve ">=" or "<=".
>
> How can I solve the following problem with LinearProgramming[]:
>
> Assuming x+2y-z>0,x+y-z>=60,y+2z>12,x>0,y>0 and z>1
>
> we want to get the minimum of 2x+3y+4z in the abrove constraints.
>
> Note that there are ">" other than">=" in the constraints we have
> given.
>
If an LP problem has a unique optimal solution then it must lie along the 
boundary of a polyhedron formed by the constraints. This implies that strictly 
less than (<) or strictly greater than (>) constraints do not fit into a LP 
problem formulation; i.e. the solution is no longer guaranteed to be along a 
boundary. This means that the function LinearProgramming is not applicable for 
"<" and ">" constraints.

I believe that you will find that if these strict constraints have any effect on 
an optimization problem, the effect will be that there is no unique solution to 
the problem --- this applies to both linear and nonlinear optimization problems. 
Thus, I suggest that you try to reformulate your problem such that it can be 
written without any strict constraints. Here is your problem reformulated with 
">" replaced by ">=".

m = {{1, 2, -1}, {1, 1, -1}, {0, 1, 2}};
b = {0, 60, 12}; s = {1, 1, 1};
c = {2, 3, 4};
bs = Transpose[{b, s}];
LinearProgramming[c, m, bs, {0, 0, 1}]

which gives {51,10,1} as the solution to,

minimize 2x + 3y + 4z
  subject to:
x + 2y -  z >= 0
x +  y -  z >= 60
      y + 2z >= 12
x >= 0, y >= 0, z >= 1

Note, two of the constraints (6 in total) in your original problem are not 
satisfied. It might useful for you to plot the polyhedron formed by these 
constraints --- graphical analysis can be enlightening.



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