Constraints - linear programming- CPLEX

Question

How do I write such constraints in CPLEX (linear-programming) ?

forall(p in P, x in X, y in Y)
  if ((remx[p,x] <= 0) OR (remy[p,y] <= 0)) then
    pbl[p,x,y] == 0 // MUST be 0
  else 
    pbl[p,x,y] == 1 OR == 0 // can be 0 or 1

where pbl is a decision variable (matrix), remx and remy is a normal matrix variable and p,x,y are indices.

I can not use if-then

Thanks,

Erwin Kalvelagen · Accepted Answer · 2016-05-23 21:28:35Z

1

I believe this cannot be done using (continuous) linear programming, but using mixed-integer programming we can use binary variables.

One way to attack this is using a bunch of inequalities, something like:

remx[p,x] <= 0 + bx[p,x]*M
remx[p,x] >= 0 - (1-bx[p,x])*M
remy[p,y] <= 0 + by[p,y]*M
remy[p,y] >= 0 - (1-by[p,y])*M
pbl[p,x,y] >= bx[p,x]+by[p,y]-1
pbl[p,x,y] <= bx[p,x]
pbl[p,x,y] <= bx[p,x]
bx[p,x],bx[p,x] in {0,1}

where M is indicating a sufficiently large number (they form a bound on remx and remy). Alternatively you can use the indicator constraints in Cplex to model implications:

bx[p,x]=0 => remx[p,x] <= 0
bx[p,x]=1 => remx[p,x] >= 0
by[p,y]=0 => remy[p,y] <= 0
by[p,y]=1 => remy[p,y] >= 0
pbl[p,x,y] = 1 => bx[p,x]+by[p,y] = 2
pbl[p,x,y] = 0 => bx[p,x]+by[p,y] <= 1

(Note: the question has changed, so these fragments are no longer 100% correct).

edited May 23, 2016 at 21:28

answered May 21, 2016 at 19:04

Erwin Kalvelagen

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3 Comments

Cristian Morales Over a year ago

It works! But I left something... pbl[p,x,y] must be 0 if the condition is true, but if not, could be 1 or 0... I mean, there are other constraints that manage this

Erwin Kalvelagen Over a year ago

One way implications are typically easier to model

Cristian Morales Over a year ago

So, Is it possible to do it without mixed-integer programming?

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