Re: Problem with Maximize and conditions.

*To*: mathgroup at smc.vnet.net*Subject*: [mg51064] Re: Problem with Maximize and conditions.*From*: "Curt Fischer" <crf3 at po.cwru.edu>*Date*: Sun, 3 Oct 2004 05:47:35 -0400 (EDT)*References*: <cjlna7$q7f$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Nacho wrote: > Hello. > > I was trying to solve a problem with Mathematica 5 and I am getting > strange results. > > The problem is: > > Minimize x+y+z, with the condition that 1/20x+y+5z==100 and x,y,z are > Integers between 1 and 98 (inclusive). > > So I use: > > Minimize[{x+y+z, 1/20 x+y+5z\[Equal]100, x \[Element] Integers, > y \[Element] Integers, z\[Element]Integers, 0<x<99,0<y<99,0<z<99}, > {x,y, > z}] > > I have copied the text using "Plain text" option, I hope it's fine. > > This returns the same expression, I suppose that Mathematica cannot > resolve it. So I use NMinimize: > > NMinimize[{x+y+z, 1/20 x+y+5z\[Equal]100, x \[Element] Integers, > y \[Element] Integers, z\[Element]Integers, 0<x<99,0<y<99,0<z<99}, > {x,y, > z}] > > Now I get a result, but rather weird... > > \!\({25.`, {x -> 1, y -> 5, z -> 1899\/100}}\) > > The minimum of x+y+z is 25 but z is 1899/100 > 1899/100 is not a Integers, and the nearest Integer, 19, doesn't > satisfy 1/20x+y+5z==100, and also x+y+z is not 25 but 24.99 > > I don't know why Mathematica has returned a Real when I specified an > Integers. I suppose that it is related to the use of NMinimize. I > suppose that it considers that 18.99 is so near of 19 that it can be > considered an Integer. > > If you remove the condition of z being an Integer, the result changes, > so it is affecting. Also, if you ask for "1899/100 e Integers" it > returns False. > > So, does anybody know how to solve this? Ideally, I would like to know > why Minimize doesn't work (so I have to use NMinimize), but in any > case, how to solve the problem. I don't know why Minimize[] and NMinimize[] are not working but the problem is small enough that an exhaustive search can find the answer easily. In[1]:= constraintmatrix= Table[Table[Table[x/20 + y + 5z,{z,1,98}],{y,1,98}],{x,1,98}]; In[2]:= feasiblespace=Position[constraintmatrix,100]; In[3]:= cost=Apply[Plus,feasiblespace,{1}]; In[4]:= feasiblespace[[#]]&/@Position[cost,Min[cost]] Out[4]= {{{20,4,19}}} Therefore, the optimal values for {x,y,z} is {20,4,19} and the minimal cost x+y+z is thus 43. The answer NMinimize was giving did not satisfy the integer constraints. In fact, because y and z must be integers, and x/20 + y + 5z must sum to an integer, we can tell right away that x must be a multiple of 20. -- Curt Fischer