Hi!
I am new to OPL and CPLEX and I'm trying to formulate an objective function where I want to maximize the number of products delivered within a time limit.

Some of my input and decision variables:
int d[S][T] = ...; // demand from site i at time t
int l = ...; // time limit in which the product needs to be at the depot after demand is available (240 minutes)
int l2 = ...; // time limit 2 (360 minutes)

dvar boolean x[S][T][V][R]; // equal to 1 if products at site i at time t are picked up by vehicle v in route r, 0 otherwise
dvar int+ w[V][R]; // arrival time of vehicle v at depot after completing route r

So when the demand becomes available, the products needs to be at the depot within 'l' minutes, otherwise penalties will occur.
If the product arrives: 
- within the limit (240 minutes) --> no penalty
- between 241 and 360 minutes --> penalty1 occurs (p1)
- after 361 minutes --> penalty2 occurs (p1)

Right now I have this objective function, but it won't work because I put a decision variable as condition in the sum.

maximize sum (i in S, t in T : w[v][r] <= t + l, v in V, r in R) d[i][t] * x[i][t][v][r]
+ sum (i in S, t in T : w[v][r] => t + l && w[v][r] <= t + l2, v in V, r in R) d[i][t] * x[i][t][v][r] * p1
+ sum (i in S, t in T : w[v][r] => t + l2, v in V, r in R) d[i][t] * x[i][t][v][r] * p2

Is something like this possible in CPLEX?
Thank you in advance!

------------------------------
Floor M
------------------------------

#DecisionOptimization

You cannot condition sums on decision variable values directly using CPLEX. You can accomplish what you want by adding more binary variables and more constraints.

Let delta[v][r] and lambda[v][r] be binary variables. Add the constraints w[v][r] <= t + l + (l2-l)*delta[v][r] + (M-l)*lambda[v][r], where M is a (constant) upper bound on how long after release the product hits the depot. Change the objective function to just the sum of c[i][t][v][r], where c is a new (nonnegative continuous) variable with the same dimensions as x. Next, add the following constraints for all combinations of i, t, v and r:

(1) c[i][t][v][r] <= d[i][t] * x[i][t][v][r]

(2) c[i][t][v][r] <= d[i][t] * (1 - (1-p)*delta[v][r] - (1-p2)*lambda[v][r])

If x is 0, (1) forces the contribution of this term to the objective to be 0. Otherwise, if the stuff is on time, the solver will set delta and lambda to 0 and get the full value d[i][t] as the objective contribution. If the stuff is mildly late, the solver will set delta to 1 and lambda to 0 and get p*d[i][t] as contribution. If it's serious late, the solver will set delta to 0 and lambda to 1, and get p2*d[i][t] as the objective contribution.


------------------------------
Paul Rubin
Professor Emeritus
Michigan State University
------------------------------

Original Message:
Sent: Wed May 20, 2020 05:10 AM
From: Floor M
Subject: Conditional sum objective function with decision variable

Hi!
I am new to OPL and CPLEX and I'm trying to formulate an objective function where I want to maximize the number of products delivered within a time limit.

Some of my input and decision variables:
int d[S][T] = ...; // demand from site i at time t
int l = ...; // time limit in which the product needs to be at the depot after demand is available (240 minutes)
int l2 = ...; // time limit 2 (360 minutes)

dvar boolean x[S][T][V][R]; // equal to 1 if products at site i at time t are picked up by vehicle v in route r, 0 otherwise
dvar int+ w[V][R]; // arrival time of vehicle v at depot after completing route r

So when the demand becomes available, the products needs to be at the depot within 'l' minutes, otherwise penalties will occur.
If the product arrives: 
- within the limit (240 minutes) --> no penalty
- between 241 and 360 minutes --> penalty1 occurs (p1)
- after 361 minutes --> penalty2 occurs (p1)

Right now I have this objective function, but it won't work because I put a decision variable as condition in the sum.

maximize sum (i in S, t in T : w[v][r] <= t + l, v in V, r in R) d[i][t] * x[i][t][v][r]
+ sum (i in S, t in T : w[v][r] => t + l && w[v][r] <= t + l2, v in V, r in R) d[i][t] * x[i][t][v][r] * p1
+ sum (i in S, t in T : w[v][r] => t + l2, v in V, r in R) d[i][t] * x[i][t][v][r] * p2

Is something like this possible in CPLEX?
Thank you in advance!

------------------------------
Floor M
------------------------------

#DecisionOptimization

Hi,

you could also with a minimal change to your model use logical constraint in the objective:

range S=1..2;
range T=1..3;
range V=1..2;
range R=1..2;

int d[s in S][t in T] = s*t; // demand from site i at time t
int l = 10; // time limit in which the product needs to be at the depot after demand is available (240 minutes)
int l2 = 10; // time limit 2 (360 minutes)

int p1=1;
int p2=2;

dvar boolean x[S][T][V][R]; // equal to 1 if products at site i at time t are picked up by vehicle v in route r, 0 otherwise
dvar int+ w[V][R]; // arrival time of vehicle v at depot after completing route r

maximize sum (i in S, t in T , v in V, r in R) ( w[v][r] <= t + l) *d[i][t] * x[i][t][v][r]
+ sum (i in S, t in T , v in V, r in R) ( (w[v][r] >= t + l) && (w[v][r] <= t + l2))*d[i][t] * x[i][t][v][r] * p1
+ sum (i in S, t in T , v in V, r in R) (w[v][r] >= t + l2)*d[i][t] * x[i][t][v][r] * p2;

subject to
{
  
}
​

works fine

------------------------------
ALEX FLEISCHER
------------------------------

Original Message:
Sent: Wed May 20, 2020 06:49 PM
From: Paul Rubin
Subject: Conditional sum objective function with decision variable

You cannot condition sums on decision variable values directly using CPLEX. You can accomplish what you want by adding more binary variables and more constraints.

Let delta[v][r] and lambda[v][r] be binary variables. Add the constraints w[v][r] <= t + l + (l2-l)*delta[v][r] + (M-l)*lambda[v][r], where M is a (constant) upper bound on how long after release the product hits the depot. Change the objective function to just the sum of c[i][t][v][r], where c is a new (nonnegative continuous) variable with the same dimensions as x. Next, add the following constraints for all combinations of i, t, v and r:

(1) c[i][t][v][r] <= d[i][t] * x[i][t][v][r]

(2) c[i][t][v][r] <= d[i][t] * (1 - (1-p)*delta[v][r] - (1-p2)*lambda[v][r])

If x is 0, (1) forces the contribution of this term to the objective to be 0. Otherwise, if the stuff is on time, the solver will set delta and lambda to 0 and get the full value d[i][t] as the objective contribution. If the stuff is mildly late, the solver will set delta to 1 and lambda to 0 and get p*d[i][t] as contribution. If it's serious late, the solver will set delta to 0 and lambda to 1, and get p2*d[i][t] as the objective contribution.


------------------------------
Paul Rubin
Professor Emeritus
Michigan State University

Original Message:
Sent: Wed May 20, 2020 05:10 AM
From: Floor M
Subject: Conditional sum objective function with decision variable

Hi!
I am new to OPL and CPLEX and I'm trying to formulate an objective function where I want to maximize the number of products delivered within a time limit.

Some of my input and decision variables:
int d[S][T] = ...; // demand from site i at time t
int l = ...; // time limit in which the product needs to be at the depot after demand is available (240 minutes)
int l2 = ...; // time limit 2 (360 minutes)

dvar boolean x[S][T][V][R]; // equal to 1 if products at site i at time t are picked up by vehicle v in route r, 0 otherwise
dvar int+ w[V][R]; // arrival time of vehicle v at depot after completing route r

So when the demand becomes available, the products needs to be at the depot within 'l' minutes, otherwise penalties will occur.
If the product arrives: 
- within the limit (240 minutes) --> no penalty
- between 241 and 360 minutes --> penalty1 occurs (p1)
- after 361 minutes --> penalty2 occurs (p1)

Right now I have this objective function, but it won't work because I put a decision variable as condition in the sum.

maximize sum (i in S, t in T : w[v][r] <= t + l, v in V, r in R) d[i][t] * x[i][t][v][r]
+ sum (i in S, t in T : w[v][r] => t + l && w[v][r] <= t + l2, v in V, r in R) d[i][t] * x[i][t][v][r] * p1
+ sum (i in S, t in T : w[v][r] => t + l2, v in V, r in R) d[i][t] * x[i][t][v][r] * p2

Is something like this possible in CPLEX?
Thank you in advance!

------------------------------
Floor M
------------------------------

#DecisionOptimization

Thank you so much for your reply, it works now! :)

------------------------------
Floor M
------------------------------

Original Message:
Sent: Thu May 21, 2020 03:29 AM
From: ALEX FLEISCHER
Subject: Conditional sum objective function with decision variable

Hi,

you could also with a minimal change to your model use logical constraint in the objective:

range S=1..2;range T=1..3;range V=1..2;range R=1..2;int d[s in S][t in T] = s*t; // demand from site i at time tint l = 10; // time limit in which the product needs to be at the depot after demand is available (240 minutes)int l2 = 10; // time limit 2 (360 minutes)int p1=1;int p2=2;dvar boolean x[S][T][V][R]; // equal to 1 if products at site i at time t are picked up by vehicle v in route r, 0 otherwisedvar int+ w[V][R]; // arrival time of vehicle v at depot after completing route rmaximize sum (i in S, t in T , v in V, r in R) ( w[v][r] <= t + l) *d[i][t] * x[i][t][v][r]+ sum (i in S, t in T , v in V, r in R) ( (w[v][r] >= t + l) && (w[v][r] <= t + l2))*d[i][t] * x[i][t][v][r] * p1+ sum (i in S, t in T , v in V, r in R) (w[v][r] >= t + l2)*d[i][t] * x[i][t][v][r] * p2;subject to{  }​

works fine

------------------------------
ALEX FLEISCHER

Original Message:
Sent: Wed May 20, 2020 06:49 PM
From: Paul Rubin
Subject: Conditional sum objective function with decision variable

You cannot condition sums on decision variable values directly using CPLEX. You can accomplish what you want by adding more binary variables and more constraints.

Let delta[v][r] and lambda[v][r] be binary variables. Add the constraints w[v][r] <= t + l + (l2-l)*delta[v][r] + (M-l)*lambda[v][r], where M is a (constant) upper bound on how long after release the product hits the depot. Change the objective function to just the sum of c[i][t][v][r], where c is a new (nonnegative continuous) variable with the same dimensions as x. Next, add the following constraints for all combinations of i, t, v and r:

(1) c[i][t][v][r] <= d[i][t] * x[i][t][v][r]

(2) c[i][t][v][r] <= d[i][t] * (1 - (1-p)*delta[v][r] - (1-p2)*lambda[v][r])

If x is 0, (1) forces the contribution of this term to the objective to be 0. Otherwise, if the stuff is on time, the solver will set delta and lambda to 0 and get the full value d[i][t] as the objective contribution. If the stuff is mildly late, the solver will set delta to 1 and lambda to 0 and get p*d[i][t] as contribution. If it's serious late, the solver will set delta to 0 and lambda to 1, and get p2*d[i][t] as the objective contribution.


------------------------------
Paul Rubin
Professor Emeritus
Michigan State University

Original Message:
Sent: Wed May 20, 2020 05:10 AM
From: Floor M
Subject: Conditional sum objective function with decision variable

Hi!
I am new to OPL and CPLEX and I'm trying to formulate an objective function where I want to maximize the number of products delivered within a time limit.

Some of my input and decision variables:
int d[S][T] = ...; // demand from site i at time t
int l = ...; // time limit in which the product needs to be at the depot after demand is available (240 minutes)
int l2 = ...; // time limit 2 (360 minutes)

dvar boolean x[S][T][V][R]; // equal to 1 if products at site i at time t are picked up by vehicle v in route r, 0 otherwise
dvar int+ w[V][R]; // arrival time of vehicle v at depot after completing route r

So when the demand becomes available, the products needs to be at the depot within 'l' minutes, otherwise penalties will occur.
If the product arrives: 
- within the limit (240 minutes) --> no penalty
- between 241 and 360 minutes --> penalty1 occurs (p1)
- after 361 minutes --> penalty2 occurs (p1)

Right now I have this objective function, but it won't work because I put a decision variable as condition in the sum.

maximize sum (i in S, t in T : w[v][r] <= t + l, v in V, r in R) d[i][t] * x[i][t][v][r]
+ sum (i in S, t in T : w[v][r] => t + l && w[v][r] <= t + l2, v in V, r in R) d[i][t] * x[i][t][v][r] * p1
+ sum (i in S, t in T : w[v][r] => t + l2, v in V, r in R) d[i][t] * x[i][t][v][r] * p2

Is something like this possible in CPLEX?
Thank you in advance!

------------------------------
Floor M
------------------------------

#DecisionOptimization