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04-10-2017 10:05 AM

Hi,

We trying to solve the next optimization problem?

We have:

- Pij, i=1÷3, j=1÷30, Pij are positive
- Bi, i=1÷3, integer positiveConstraints:

The searching result is matrix of 3 x 30 of binary values Xij with next conditions:

- For each j =1÷30, Sum (by index of i=1÷3)Xij=1
- For each i =1÷3, Sum by index of (j=1÷3o)Xij≤Bi

Objective: Optimize:

- Maximize (Sum (by index of i=1÷3) Sum (by index of j=1÷30) Pij *Xij)Is there another predefined subroutine in SAS/IML which can be used to solve this problem? We haven’t the SAS/OR.

The suggested solution from Rick_SAS was to convert this to linear optimization programing.

You can see here the solution.

Linear optimization in SAS/IML

In real situation we for j index we have 30000, not only 30. The matrix became too large. It is not possible to solve it by SAS/IML. The limitation there is 200 constraints, which is too few.

We haven’t SAS/OR yet.

My question concerns the limitation of constraints and the size of matrix in SAS/OR? Is it possible to work with matrix size of 90000 x 30000. How many constraints there can have?

Is it possible to solve this task with another procedure, not with linear programing in SAS/OR?

Thanks,

Boriana

Accepted Solutions

Solution

04-12-2017
03:50 AM

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04-10-2017 12:05 PM

You can formulate and solve the problem with PROC OPTMODEL in SAS/OR:

```
proc optmodel;
set ISET = 1..3;
set JSET = 1..30000;
num p {ISET, JSET} = rand('UNIFORM');
num b {ISET} = card(JSET)*rand('UNIFORM');
var X {ISET, JSET} binary;
max Z = sum {i in ISET, j in JSET} p[i,j] * X[i,j];
con Con1 {j in JSET}:
sum {i in ISET} X[i,j] = 1;
con Con2 {i in ISET}:
sum {j in JSET} X[i,j] <= b[i];
/* call (default) MILP solver */
solve;
/* call LP solver with (default) dual simplex algorithm */
solve with lp relaxint;
/* call LP solver with network simplex algorithm */
solve with lp relaxint / algorithm=ns;
quit;
```

Because the problem has a pure network structure, the third solve performs the best, solving in less than a second. Due to total unimodularity of the resulting constraint matrix, you can relax the integrality of x and an optimal solution will automatically take integer values.

You can also model the problem in terms of a bipartite directed network, where the left side consists of the i nodes, each with a supply of at most b[i], and the right side consists of the j nodes, each with a demand of exactly 1. The (binary) variable x[i,j] represents the flow from node i to node j. You can negate the link weights (because you want to maximize) and call the minimum-cost network flow algorithm either via the network solver in PROC OPTMODEL or via PROC OPTNET, also in SAS/OR.

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Solution

04-12-2017
03:50 AM

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04-10-2017 12:05 PM

You can formulate and solve the problem with PROC OPTMODEL in SAS/OR:

```
proc optmodel;
set ISET = 1..3;
set JSET = 1..30000;
num p {ISET, JSET} = rand('UNIFORM');
num b {ISET} = card(JSET)*rand('UNIFORM');
var X {ISET, JSET} binary;
max Z = sum {i in ISET, j in JSET} p[i,j] * X[i,j];
con Con1 {j in JSET}:
sum {i in ISET} X[i,j] = 1;
con Con2 {i in ISET}:
sum {j in JSET} X[i,j] <= b[i];
/* call (default) MILP solver */
solve;
/* call LP solver with (default) dual simplex algorithm */
solve with lp relaxint;
/* call LP solver with network simplex algorithm */
solve with lp relaxint / algorithm=ns;
quit;
```

Because the problem has a pure network structure, the third solve performs the best, solving in less than a second. Due to total unimodularity of the resulting constraint matrix, you can relax the integrality of x and an optimal solution will automatically take integer values.

You can also model the problem in terms of a bipartite directed network, where the left side consists of the i nodes, each with a supply of at most b[i], and the right side consists of the j nodes, each with a demand of exactly 1. The (binary) variable x[i,j] represents the flow from node i to node j. You can negate the link weights (because you want to maximize) and call the minimum-cost network flow algorithm either via the network solver in PROC OPTMODEL or via PROC OPTNET, also in SAS/OR.

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04-11-2017 04:58 AM

Do you know if there are any limitation of this procedure? For the number of constraints, for size of datasets? Does it depend from RAM of the computer?

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04-11-2017 10:04 AM

There is a hard limit of 2^31 - 1 = 2,147,483,647 variables, constraints, or nonzero coefficients. Other than that, the only limit depends on the amount of memory you have provided to SAS.

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04-11-2017 09:56 AM

Or you could perform Genetic Algorithm in IML .