[AMPL 24879] Searching for the starting date of the project using a binary variable Act(j,t)

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1

Hi,

I would like to know what’s the problem with all the statements that I tried to use in order to search about the first date t in T where Act[j,t] equals to 1.

In fact, Act[i,j] is a binary variable which equals to 1 is the project j is executed in the period t and 0 otherwise.

I am trying to express the starting and finishing date of my model using the variable Act[i,j] where :

  • Start = the first date t where Act[j,t]=1
  • Finish= the last date t where Act[j,t]=1

For Finish date, I used the following statement, and it is working :

subject to FinContrainte{j in P}:

Fin[j] = max {t in T} (t * Act[j, t]);

However, for the start date none of the following statement worked :

subject to DebutContrainte{j in P}:
Debut[j] = ord(t in T: Act[j, t] > 0, t); #Here it said syntax error

subject to DebutContrainte{j in P}:
Debut[j] = ord(t in T, (Act[j, t] = 1)); #Here it said syntax error as well

subject to FinContrainte{j in P}:

Debut[j] = min {t in T} (t * Act[j, t]); #Here it just presents Debut = 0 as there is no condition on Act(j,t)

subject to DebutContrainte{j in P}:
Debut[j] = ord(t, (t in T)$(Act[j, t] = 1)); #Here it said syntax error as well

For the following statement :

subject to DebutContrainte{j in P}:

Debut[j] = min {t in T: Act[j, t] = 1} t; it shows variable in such-that part of set specification

context: Debut[j] = min {t in T: Act[j, t] = >>> 1} <<< t;

Thanks in advance!!!

2

Hi,

Debut[j] = min {t in T} (t * Act[j, t]);

would work if you had constraints on Act preventing them to become 1 before actual start.

Depending on the difficulty of your instances, you might want a more efficient formulation using auxiliary variables S0[j, t] (“start on or before) and S1 (start on) and constraints

S0[j, 1] = Act[j, 1];
S0[j, t] >= Act[j, t];
S0[j, t] >= S0[j, t-1];
S0 jT = 1;

S1 j1 = S0 j1;
S1 jt = S0 jt - S0 j,t-1;
S1 jt <= Act jt;

Depending on your model, you might need only a part of these. For example, when minimising the sum of completion times, “end on or after” variables can be maximised.

3

EDIT.

You are right, the start time cannot be constrained by

Debut[j] = min {t in T} (t * Act[j, t]);

because any time slot with Act[j, t]=0 sets it to 0. Instead, go as follows:

Debut[j] = T - max {t in T} ((T-t) * Act[j, t]);

For the alternative modeling with binary variables S0/S1, the start time can be computed as

Debut[j] = sum {t in T} (t * S1[j, t]);
4

Good afternoon,

The following one works perfectly :
Debut[j] = T - max {t in T} ((T-t) * Act[j, t]);

Many thanks!!!

5

Good afternoon,

The following one works perfectly : Debut[j] = T - max {t in T} ((T-t) * Act[j, t]);

Thank you so much