[R] linear programming in R | limits to what it can do, or my mistake?

[Evan Cooch]

Question for 'experts' in LP using R (using the lpSolve package, say) -- 
which does not apply to me for the sort of problem I describe below.
I've run any number of LP's using lpSolve in R, but all of them to date 
have objective and constraint functions that both contain the same 
variables. This lets you set up a LHS and RHS matrix/vector that are 
symmetrical.

But, for a problem a student posed in class, I'm stuck with how to do it 
in R, if its even possible (its trivial in Maxima, Maple...even using 
Solver in Excel, but I haven't been remotely successful in getting 
anything to work in R).

Suppose you have a production system that at 4 sequential time steps 
generate 640, 825, 580, and 925 units. At each time step, you need to 
decide how many of those units need to be 'quality control' (QC) checked 
in some fashion, subject to some constraints.

  --> at no point in time can the number of units in the system be >1000
  --> at the end of the production cycle, there can be no units left
  --> 'QC checking' costs money, varying as a function of the time step 
-- 35, 55, 50 and 65 for each unit, for each time step in turn.

Objective is to minimize total cost. The total cost objective function 
is trivial. Let p1 = number sent out time step 1, p2 number sent out at 
time step 3, and so on. So, total cost function we want to minimize is 
simply

   cost=(35*p1)+(55*p2)+(50*p3)+(65*p4)

where p1+p2+p3+p4=(640+825+580+925)=2970 (i.e., all the products get 
checked). The question is, what number do you send out at each time step 
to minimize cost?

Where I get hung up in R is the fact that if I let t(i) be the number of 
products at each time step, then

     t1=640,
     t2=t1-p1+825
     t3=t2-p2+580
     t4=t3-p3+925

such that t1+t2+t3+t4=2970 (as it must), with additional constraints being

   p1<=t1, p2<=t2, p3<=t3, p4<=t4, {t1..t4}<=1000, and t4-p4=0.

There may be algebraic ways to reduce the number of functions needed to 
describe the constraints, but I can't for the life of me see how I can 
create a coefficient matrix (typically, the LHS) since each line of said 
matrix, which corresponds to the constraints, needs to be a function of 
the unknowns in the objective function -- being, p1, p2, p3 and p4.

In Maple (for example), this is trivial:

      cost:=35*p10+55*p12+50*p14+65*p16;
cnsts:={t10=640,t12=t10-p10+825,t14=t12-p12+580,t16=t14-p14+925,t16-p16=0,p10<=t10,p12<=t12,p14<=t14,p16<=t16,t10<=1000,t12<=1000,t14<=1000,t16<=1000};
      Minimize(cost,cnsts,assume={nonnegative});

which yields (correctly):

p1=640, p2=405, p3=1000, p4=925

for minimized cost of 154800.

Took only a minute to also set this up in Maxima, and using Solver in 
Excel. But danged if I can suss out any way to do this in R.

Pointers to the obvious welcome.
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