The OPTMILP Procedure

This example illustrates a model in an MPS-format SAS data set. This data set is passed to PROC OPTMILP, and a solution is found.

Consider a scenario where you have a container with a set of limiting attributes (volume V and weight W) and a set I of items that you want to pack. Each item type i has a certain value , a volume , and a weight . You must choose at most four items of each type so that the total value is maximized and all the chosen items fit into the container. Let be the number of items of type i to be included in the container. This model can be formulated as the following integer linear program:

Constraint (volume_con) enforces the volume capacity limit, while constraint (weight_con) enforces the weight capacity limit. An instance of this problem can be saved in an MPS-format SAS data set by using the following code:

data ex1data;
    input field1 $ field2 $ field3 $ field4 field5 $ field6;
    datalines;
NAME       .             ex1data           .     .                 .
ROWS       .             .                 .     .                 .
MAX        z             .                 .     .                 .
L          volume_con    .                 .     .                 .
L          weight_con    .                 .     .                 .
COLUMNS    .             .                 .     .                 .
.          .MRK0         'MARKER'          .     'INTORG'          .
.          x[1]          z                 1     volume_con       10
.          x[1]          weight_con       12     .                 .
.          x[2]          z                 2     volume_con      300
.          x[2]          weight_con       15     .                 .
.          x[3]          z                 3     volume_con      250
.          x[3]          weight_con       72     .                 .
.          x[4]          z                 4     volume_con      610
.          x[4]          weight_con      100     .                 .
.          x[5]          z                 5     volume_con      500
.          x[5]          weight_con      223     .                 .
.          x[6]          z                 6     volume_con      120
.          x[6]          weight_con       16     .                 .
.          x[7]          z                 7     volume_con       45
.          x[7]          weight_con       73     .                 .
.          x[8]          z                 8     volume_con      100
.          x[8]          weight_con       12     .                 .
.          x[9]          z                 9     volume_con      200
.          x[9]          weight_con      200     .                 .
.          x[10]         z                10     volume_con       61
.          x[10]         weight_con      110     .                 .
.          .MRK1         'MARKER'          .     'INTEND'          .
RHS        .             .                 .     .                 .
.          .RHS.         volume_con     1000     .                 .
.          .RHS.         weight_con      500     .                 .
BOUNDS     .             .                 .     .                 .
UP         .BOUNDS.      x[1]              4     .                 .
UP         .BOUNDS.      x[2]              4     .                 .
UP         .BOUNDS.      x[3]              4     .                 .
UP         .BOUNDS.      x[4]              4     .                 .
UP         .BOUNDS.      x[5]              4     .                 .
UP         .BOUNDS.      x[6]              4     .                 .
UP         .BOUNDS.      x[7]              4     .                 .
UP         .BOUNDS.      x[8]              4     .                 .
UP         .BOUNDS.      x[9]              4     .                 .
UP         .BOUNDS.      x[10]             4     .                 .
ENDATA     .             .                 .     .                 .
;

In the COLUMNS section of this data set, the name of the objective is z, and the objective coefficients appear in field4. The coefficients of (volume_con) appear in field6. The coefficients of (weight_con) appear in field4. In the RHS section, the bounds V and W appear in field4. This problem can be solved by using the following statements to call the OPTMILP procedure:

proc optmodel;
    num nItems          = 10;
    num volume_capacity = 1000;
    num weight_capacity = 500;
    set<num> Items = {1..nItems};
    num value{Items}  = [1,2,3,4,5,6,7,8,9,10];
    num volume{Items} = [10, 300, 250, 610, 500, 120, 45, 100, 200, 61 ];
    num weight{Items} = [12,  15,  72, 100, 223,  16, 73,  12, 200, 110];
    var x{Items} integer >= 0 <= 4;
    max z = sum{i in Items} value[i] * x[i];
    con volume_con: sum{i in Items} volume[i] * x[i] <= volume_capacity;
    con weight_con: sum{i in Items} weight[i] * x[i] <= weight_capacity;
    save mps ex1data;
  quit;
run;
proc optmilp data=ex1data primalout=ex1soln;
run;

The progress of the solver is shown in Output 12.1.1.

The data set ex1soln is shown in Output 12.1.2.

The optimal solution is , and , with a total value of 87. From this solution, you can compute the total volume used, which is 988 (); the total weight used is 499 (). The problem summary and solution summary are shown in Output 12.1.3.

proc optmodel;
    num nItems          = 10;
    num volume_capacity = 1000;
    num weight_capacity = 500;
    set<num> Items = {1..nItems};
    num value{Items}  = [1,2,3,4,5,6,7,8,9,10];
    num volume{Items} = [10, 300, 250, 610, 500, 120, 45, 100, 200, 61 ];
    num weight{Items} = [12,  15,  72, 100, 223,  16, 73,  12, 200, 110];
    var x{Items} integer >= 0 <= 4;
    max z = sum{i in Items} value[i] * x[i];
    con volume_con: sum{i in Items} volume[i] * x[i] <= volume_capacity;
    con weight_con: sum{i in Items} weight[i] * x[i] <= weight_capacity;
    save mps ex1data;
  quit;
run;
proc optmilp data=ex1data primalout=ex1soln;
title ' ';
run;