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

[Jinsong Zhao]

On 2024/1/30 20:00, Martin Becker wrote:
> Apart from the fact that the statement "such that t1+t2+t3+t4=2970 (as 
> it must)" is not correct, the LP can be implemented as follows:
>
I was confused by "such that t1+t2+t3+t4=2970 (as it must)", otherwise, 
I also get the same solution.
> library(lpSolve)
> LHS <- rbind(
> c(0,0,0,0, 1, 0, 0,0),
> c(1,0,0,0,-1, 1, 0,0),
> c(0,1,0,0, 0,-1, 1,0),
> c(0,0,1,0, 0, 0,-1,1),
> cbind(-diag(4),diag(4)),
> c(0,0,0,0,0,1,0,0),
> c(0,0,0,0,0,0,1,0),
> c(0,0,0,0,0,0,0,1)
> )
>
> RHS <- c(640,825,580,925,0,0,0,0,1000,1000,1000)
>
> DIR <- c(rep("==",4),rep(">=",3),"=",rep("<=",3))
>
> OBJ <- c(35,55,50,65,0,0,0,0)
>
> lp("min",OBJ,LHS,DIR,RHS)
>
> Best,
> Martin
>
>
> Am 29.01.24 um 22:28 schrieb 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.
>>     [[alternative HTML version deleted]]
>>
>>
>