Re: LinearProgramming[]
- To: mathgroup at smc.vnet.net
- Subject: [mg123554] Re: LinearProgramming[]
- From: DrMajorBob <btreat1 at austin.rr.com>
- Date: Sun, 11 Dec 2011 03:46:18 -0500 (EST)
- Delivered-to: l-mathgroup@mail-archive0.wolfram.com
- References: <201112091058.FAA03920@smc.vnet.net>
- Reply-to: drmajorbob at yahoo.com
> 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.
No OPTIMAL solution, actually... unique or not.
Bobby
On Sat, 10 Dec 2011 06:28:53 -0600, Virgil Stokes <vs at it.uu.se> wrote:
> On 09-Dec-2011 11:58, é�â?? 厚 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.
>
--
DrMajorBob at yahoo.com
- References:
- LinearProgramming[]
- From: é å <shlwell1988@gmail.com>
- LinearProgramming[]