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Example 4.6: Adding Side Constraints, Using a Warm Start

The manufacturer of Gizmo chips, which are parts needed to make televisions, can supply only 2600 chips to factory 1 and 3750 chips to factory 2 in time for production in each of the months 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 ARCOUT= 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 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 NETFLOW 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' .

title 'Adding Side Constraints and Using a Warm Start';
title2 'Production Planning/Inventory/Distribution';
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 NETFLOW run. The set used depends on which cost information the arcs are to have and whether a warm start is to be used.

ARCDATA=arc0    NODEDATA=node0
ARCDATA=arc1    NODEDATA=node2
ARCDATA=arc2    NODEDATA=node2
ARCDATA=arc3    NODEDATA=node3

arc0, node0, arc1, and node2 were created in Example 4.4. The first two data sets are the original input data sets. arc1 and node2 were the ARCOUT= and NODEOUT= data sets of a PROC NETFLOW run with FUTURE1 specified. Now, if you use arc1 and node2 as the ARCDATA= data set and NODEDATA= data set in a PROC NETFLOW run, you can specify WARM, as these data sets contain additional information describing a warm start.

In Example 4.5, arc2 was created by modifying arc1 to reflect different arc costs. arc2 and node2 can also be used as the ARCDATA= and NODEDATA= data sets in a PROC NETFLOW run. Again, specify WARM, as these data sets contain additional information describing a warm start, This start, however, contains the optimal basis using the original costs.

If you are going to continue optimization using the changed arc costs, it is probably best to use arc3 and node3 as the ARCDATA= and NODEDATA= data sets. These data sets, created in Example 4.6 by PROC NETFLOW when the FUTURE1 option was specified, contain an optimal basis that can be used as a warm start.

PROC NETFLOW is used to find the changed cost network solution that obeys the chip limit and backorder side constraints. The FUTURE2 option is specified in case further processing is required. 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 netflow
      nodedata=node3 arcdata=arc3 warm
      condata=con3 sparsecondata rhsobs='CHIP/BO LIMIT'
      future2 dualout=dual4 conout=con4;
         id diagonal factory key_id mth_made;
   proc print data=con4;
      sum _fcost_;
   proc print data=dual4;

The following messages appear on the SAS log:

NOTE: The following 3 variables in ARCDATA do not belong to
      any SAS variable list. These will be ignored.
      OLDCOST
      OLDFC
      OLDFLOW
NOTE: The following 1 variables in NODEDATA do not belong to 
      any SAS variable list. These will be ignored.
      _DUAL_
NOTE: Number of nodes= 21 .
NOTE: Number of supply nodes= 4 .
NOTE: Number of demand nodes= 5 .
NOTE: The greater of total supply and total demand= 4350 .
NOTE: Number of iterations performed (neglecting any 
      constraints)= 1 .
NOTE: Of these, 0 were degenerate.
NOTE: Optimum (neglecting any constraints) found.
NOTE: Minimal total cost= -1285086.45 .
NOTE: Number of <= side constraints= 5 .
NOTE: Number of == side constraints= 0 .
NOTE: Number of >= side constraints= 0 .
NOTE: Number of arc and nonarc variable side constraint 
      coefficients= 16 .
NOTE: Number of iterations, optimizing with constraints= 10 .
NOTE: Of these, 0 were degenerate.
NOTE: Optimum reached.
NOTE: Minimal total cost= -1282708.625 .
NOTE: The data set WORK.CON4 has 68 observations and 18 
      variables.
NOTE: The data set WORK.DUAL4 has 27 observations and 14 
      variables.

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