Example 4.3 Adding Side Constraints

The manufacturer of Gizmo chips, which are parts needed to make televisions, can supply only 2,600 chips to factory 1 and 3,750 chips to factory 2 in time for production in each of the months of March and April. However, Gizmo chips will not be in short supply in May. Three chips are required to make each 19-inch TV while the 25-inch TVs require four chips each. To limit the production of televisions produced at factory 1 in March so that the TVs have the correct number of chips, a side constraint called FACT1 MAR GIZMO is used. The form of this constraint is

   3 * prod f1 19 mar   +   4 * prod f1 25 mar   <=   2600

prod f1 19 mar is the name of the arc directed from the node fact1_1 toward node f1_mar_1 and, in the previous constraint, designates the flow assigned to this arc. The ARCDATA= and CONOUT= data sets have arc names in a variable called _name_.

The other side constraints (shown below) are called FACT2 MAR GIZMO, FACT1 APL GIZMO, and FACT2 APL GIZMO.

   3 * prod f2 19 mar   +   4 * prod f2 25 mar   <=   3750
   3 * prod f1 19 apl   +   4 * prod f1 25 apl   <=   2600
   3 * prod f2 19 apl   +   4 * prod f2 25 apl   <=   3750

To maintain customer goodwill, the total number of backorders is not to exceed 50 sets. The side constraint TOTAL BACKORDER that models this restriction is

   back f1 19 apl  +  back f1 25 apl  +  
   back f2 19 apl  +  back f2 25 apl  + 
   back f1 19 may  +  back f1 25 may  +
   back f2 19 may  +  back f2 25 may   <=   50

The sparse CONDATA= data set format is used. All side constraints are of less than or equal type. Because this is the default type value for the DEFCONTYPE= option, type information is not necessary in the following CONDATA=con3. Also, DEFCONTYPE= <= does not have to be specified in the PROC INTPOINT statement that follows. Notice that the _column_ variable value CHIP/BO LIMIT indicates that an observation of the con3 data set contains rhs information. Therefore, specify RHSOBS='CHIP/BO LIMIT'

title2 'Adding Side Constraints';
data con3;
   input _column_ &$14. _row_ &$15. _coef_ ;
   datalines;
prod f1 19 mar  FACT1 MAR GIZMO  3
prod f1 25 mar  FACT1 MAR GIZMO  4
CHIP/BO LIMIT   FACT1 MAR GIZMO  2600
prod f2 19 mar  FACT2 MAR GIZMO  3
prod f2 25 mar  FACT2 MAR GIZMO  4
CHIP/BO LIMIT   FACT2 MAR GIZMO  3750
prod f1 19 apl  FACT1 APL GIZMO  3
prod f1 25 apl  FACT1 APL GIZMO  4
CHIP/BO LIMIT   FACT1 APL GIZMO  2600
prod f2 19 apl  FACT2 APL GIZMO  3
prod f2 25 apl  FACT2 APL GIZMO  4
CHIP/BO LIMIT   FACT2 APL GIZMO  3750
back f1 19 apl  TOTAL BACKORDER  1
back f1 25 apl  TOTAL BACKORDER  1
back f2 19 apl  TOTAL BACKORDER  1
back f2 25 apl  TOTAL BACKORDER  1
back f1 19 may  TOTAL BACKORDER  1
back f1 25 may  TOTAL BACKORDER  1
back f2 19 may  TOTAL BACKORDER  1
back f2 25 may  TOTAL BACKORDER  1
CHIP/BO LIMIT   TOTAL BACKORDER  50
;

The four pairs of data sets that follow can be used as ARCDATA= and NODEDATA= data sets in the following PROC INTPOINT run. The set used depends on which cost information the arcs are to have.

   ARCDATA=arc0    NODEDATA=node0
   ARCDATA=arc1    NODEDATA=node0
   ARCDATA=arc2    NODEDATA=node0
   ARCDATA=arc3    NODEDATA=node0

arc0, node0, and arc1 were created in Example 4.1. The first two data sets are the original input data sets.

In the previous example, arc2 was created by modifying arc1 to reflect different arc costs. arc2 and node0 can also be used as the ARCDATA= and NODEDATA= data sets in a PROC INTPOINT run.

If you are going to continue optimization using the changed arc costs, it is probably best to use arc3 and node0 as the ARCDATA= and NODEDATA= data sets.

PROC INTPOINT is used to find the changed cost network solution that obeys the chip limit and backorder side constraints. An explicit ID list has also been specified so that the variables oldcost, oldfc, and oldflow do not appear in the subsequent output data sets:

proc intpoint
   bytes=1000000
   printlevel2=2
   nodedata=node0 arcdata=arc3
   condata=con3 sparsecondata rhsobs='CHIP/BO LIMIT'
   conout=arc4;
   id diagonal factory key_id mth_made;
   run;

proc print data=arc4;
   var _tail_ _head_ _cost_ _capac_ _lo_ _flow_ _fcost_;
       /* to get this variable order */
   sum _fcost_;
   run;

The following messages appear on the SAS log: