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 The NETFLOW Procedure

## 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.
```

Output 4.6.1: CONOUT=CON4

 Adding Side Constraints and Using a Warm Start Production Planning/Inventory/Distribution

 Obs _tail_ _head_ _cost_ _capac_ _lo_ _name_ _SUPPLY_ _DEMAND_ _FLOW_ _FCOST_ _RCOST_ _ANUMB_ _TNUMB_ _STATUS_ diagonal factory key_id mth_made 1 fact1_1 _EXCESS_ 0.00 99999999 0 1000 200 5.000 0.00 . 65 1 KEY_ARC BASIC . . 2 fact2_1 _EXCESS_ 0.00 99999999 0 850 200 45.000 0.00 . 66 10 KEY_ARC BASIC . . 3 fact1_2 _EXCESS_ 0.00 99999999 0 1000 200 0.000 0.00 30.187 67 11 LOWERBD NONBASIC . . 4 fact2_2 _EXCESS_ 0.00 99999999 0 1500 200 150.000 0.00 . 68 20 KEY_ARC BASIC . . 5 fact1_1 f1_apr_1 78.60 600 50 prod f1 19 apl 1000 . 533.333 41920.00 . 4 1 KEY_ARC BASIC 19 1 production April 6 f1_mar_1 f1_apr_1 15.00 50 0 . . 0.000 0.00 63.650 5 2 LOWERBD NONBASIC 19 1 storage March 7 f1_may_1 f1_apr_1 33.60 20 0 back f1 19 may . . 0.000 0.00 47.020 6 4 LOWERBD NONBASIC 19 1 backorder May 8 f2_apr_1 f1_apr_1 11.00 40 0 . . 0.000 0.00 22.000 7 6 LOWERBD NONBASIC 19 . f2_to_1 April 9 fact1_2 f1_apr_2 174.50 550 50 prod f1 25 apl 1000 . 250.000 43625.00 . 36 11 KEY_ARC BASIC 25 1 production April 10 f1_mar_2 f1_apr_2 20.00 40 0 . . 0.000 0.00 94.210 37 12 LOWERBD NONBASIC 25 1 storage March 11 f1_may_2 f1_apr_2 49.20 15 0 back f1 25 may . . 0.000 0.00 . 38 14 NONKEY ARC BASIC 25 1 backorder May 12 f2_apr_2 f1_apr_2 21.00 25 0 . . 0.000 0.00 30.510 39 16 LOWERBD NONBASIC 25 . f2_to_1 April 13 fact1_1 f1_mar_1 127.90 500 50 prod f1 19 mar 1000 . 333.333 42633.33 . 1 1 KEY_ARC BASIC 19 1 production March 14 f1_apr_1 f1_mar_1 33.60 20 0 back f1 19 apl . . 20.000 672.00 -7.630 2 3 UPPERBD NONBASIC 19 1 backorder April 15 f2_mar_1 f1_mar_1 10.00 40 0 . . 40.000 400.00 -34.750 3 5 UPPERBD NONBASIC 19 . f2_to_1 March 16 fact1_2 f1_mar_2 217.90 400 40 prod f1 25 mar 1000 . 400.000 87160.00 -31.677 33 11 UPPERBD NONBASIC 25 1 production March 17 f1_apr_2 f1_mar_2 38.40 30 0 back f1 25 apl . . 30.000 1152.00 -28.390 34 13 UPPERBD NONBASIC 25 1 backorder April 18 f2_mar_2 f1_mar_2 20.00 25 0 . . 25.000 500.00 -61.060 35 15 UPPERBD NONBASIC 25 . f2_to_1 March 19 fact1_1 f1_may_1 90.10 400 50 1000 . 128.333 11562.83 . 8 1 KEY_ARC BASIC 19 1 production May 20 f1_apr_1 f1_may_1 12.00 50 0 . . 0.000 0.00 6.000 9 3 LOWERBD NONBASIC 19 1 storage April 21 f2_may_1 f1_may_1 13.00 40 0 . . 0.000 0.00 29.000 10 7 LOWERBD NONBASIC 19 . f2_to_1 May 22 fact1_2 f1_may_2 113.30 350 40 1000 . 350.000 39655.00 -11.913 40 11 UPPERBD NONBASIC 25 1 production May 23 f1_apr_2 f1_may_2 18.00 40 0 . . 0.000 0.00 74.620 41 13 LOWERBD NONBASIC 25 1 storage April 24 f2_may_2 f1_may_2 13.00 25 0 . . 0.000 0.00 39.000 42 17 LOWERBD NONBASIC 25 . f2_to_1 May 25 f1_apr_1 f2_apr_1 11.00 99999999 0 . . 13.333 146.67 . 14 3 KEY_ARC BASIC 19 . f1_to_2 April 26 fact2_1 f2_apr_1 62.40 480 35 prod f2 19 apl 850 . 480.000 29952.00 -14.077 15 10 UPPERBD NONBASIC 19 2 production April 27 f2_mar_1 f2_apr_1 18.00 30 0 . . 0.000 0.00 10.900 16 5 LOWERBD NONBASIC 19 2 storage March 28 f2_may_1 f2_apr_1 30.00 15 0 back f2 19 may . . 0.000 0.00 48.420 17 7 LOWERBD NONBASIC 19 2 backorder May 29 f1_apr_2 f2_apr_2 23.00 99999999 0 . . 0.000 0.00 13.490 46 13 LOWERBD NONBASIC 25 . f1_to_2 April 30 fact2_2 f2_apr_2 196.70 680 35 prod f2 25 apl 1500 . 577.500 113594.25 . 47 20 KEY_ARC BASIC 25 2 production April 31 f2_mar_2 f2_apr_2 28.00 50 0 . . 0.000 0.00 11.640 48 15 LOWERBD NONBASIC 25 2 storage March 32 f2_may_2 f2_apr_2 64.80 15 0 back f2 25 may . . 0.000 0.00 32.090 49 17 LOWERBD NONBASIC 25 2 backorder May 33 f1_mar_1 f2_mar_1 11.00 99999999 0 . . 0.000 0.00 55.750 11 2 LOWERBD NONBASIC 19 . f1_to_2 March 34 fact2_1 f2_mar_1 88.00 450 35 prod f2 19 mar 850 . 290.000 25520.00 . 12 10 KEY_ARC BASIC 19 2 production March 35 f2_apr_1 f2_mar_1 20.40 15 0 back f2 19 apl . . 0.000 0.00 34.920 13 6 LOWERBD NONBASIC 19 2 backorder April 36 f1_mar_2 f2_mar_2 23.00 99999999 0 . . 0.000 0.00 104.060 43 12 LOWERBD NONBASIC 25 . f1_to_2 March 37 fact2_2 f2_mar_2 182.00 650 35 prod f2 25 mar 1500 . 650.000 118300.00 -23.170 44 20 UPPERBD NONBASIC 25 2 production March 38 f2_apr_2 f2_mar_2 37.20 15 0 back f2 25 apl . . 0.000 0.00 60.980 45 16 LOWERBD NONBASIC 25 2 backorder April 39 f1_may_1 f2_may_1 16.00 99999999 0 . . 115.000 1840.00 . 18 4 KEY_ARC BASIC 19 . f1_to_2 May 40 fact2_1 f2_may_1 128.80 250 35 850 . 35.000 4508.00 22.700 19 10 LOWERBD NONBASIC 19 2 production May 41 f2_apr_1 f2_may_1 20.00 30 0 . . 0.000 0.00 9.000 20 6 LOWERBD NONBASIC 19 2 storage April 42 f1_may_2 f2_may_2 26.00 99999999 0 . . 350.000 9100.00 . 50 14 KEY_ARC BASIC 25 . f1_to_2 May 43 fact2_2 f2_may_2 181.40 550 35 1500 . 122.500 22221.50 . 51 20 NONKEY ARC BASIC 25 2 production May 44 f2_apr_2 f2_may_2 38.00 50 0 . . 0.000 0.00 78.130 52 16 LOWERBD NONBASIC 25 2 storage April 45 f1_mar_1 shop1_1 -327.65 250 0 . 900 143.333 -46963.17 . 21 2 KEY_ARC BASIC 19 1 sales March 46 f1_apr_1 shop1_1 -300.00 250 0 . 900 250.000 -75000.00 -21.000 22 3 UPPERBD NONBASIC 19 1 sales April 47 f1_may_1 shop1_1 -285.00 250 0 . 900 13.333 -3800.00 . 23 4 NONKEY ARC BASIC 19 1 sales May 48 f2_mar_1 shop1_1 -297.40 250 0 . 900 250.000 -74350.00 -14.500 24 5 UPPERBD NONBASIC 19 2 sales March 49 f2_apr_1 shop1_1 -290.00 250 0 . 900 243.333 -70566.67 . 25 6 NONKEY ARC BASIC 19 2 sales April 50 f2_may_1 shop1_1 -292.00 250 0 . 900 0.000 0.00 9.000 26 7 LOWERBD NONBASIC 19 2 sales May 51 f1_mar_2 shop1_2 -559.76 99999999 0 . 900 0.000 0.00 47.130 53 12 LOWERBD NONBASIC 25 1 sales March 52 f1_apr_2 shop1_2 -524.28 99999999 0 . 900 0.000 0.00 8.400 54 13 LOWERBD NONBASIC 25 1 sales April 53 f1_may_2 shop1_2 -475.02 99999999 0 . 900 0.000 0.00 1.040 55 14 LOWERBD NONBASIC 25 1 sales May 54 f2_mar_2 shop1_2 -567.83 500 0 . 900 500.000 -283915.00 -42.000 56 15 UPPERBD NONBASIC 25 2 sales March 55 f2_apr_2 shop1_2 -542.19 500 0 . 900 400.000 -216876.00 . 57 16 KEY_ARC BASIC 25 2 sales April 56 f2_may_2 shop1_2 -491.56 500 0 . 900 0.000 0.00 10.500 58 17 LOWERBD NONBASIC 25 2 sales May 57 f1_mar_1 shop2_1 -362.74 250 0 . 900 250.000 -90685.00 -37.090 27 2 UPPERBD NONBASIC 19 1 sales March 58 f1_apr_1 shop2_1 -300.00 250 0 . 900 250.000 -75000.00 -23.000 28 3 UPPERBD NONBASIC 19 1 sales April 59 f1_may_1 shop2_1 -245.00 250 0 . 900 0.000 0.00 38.000 29 4 LOWERBD NONBASIC 19 1 sales May 60 f2_mar_1 shop2_1 -272.70 250 0 . 900 0.000 0.00 8.200 30 5 LOWERBD NONBASIC 19 2 sales March 61 f2_apr_1 shop2_1 -312.00 250 0 . 900 250.000 -78000.00 -24.000 31 6 UPPERBD NONBASIC 19 2 sales April 62 f2_may_1 shop2_1 -299.00 250 0 . 900 150.000 -44850.00 . 32 7 KEY_ARC BASIC 19 2 sales May 63 f1_mar_2 shop2_2 -623.89 99999999 0 . 1450 455.000 -283869.95 . 59 12 KEY_ARC BASIC 25 1 sales March 64 f1_apr_2 shop2_2 -549.68 99999999 0 . 1450 220.000 -120929.60 . 60 13 KEY_ARC BASIC 25 1 sales April 65 f1_may_2 shop2_2 -460.00 99999999 0 . 1450 0.000 0.00 33.060 61 14 LOWERBD NONBASIC 25 1 sales May 66 f2_mar_2 shop2_2 -542.83 500 0 . 1450 125.000 -67853.75 . 62 15 KEY_ARC BASIC 25 2 sales March 67 f2_apr_2 shop2_2 -559.19 500 0 . 1450 177.500 -99256.23 . 63 16 KEY_ARC BASIC 25 2 sales April 68 f2_may_2 shop2_2 -519.06 500 0 . 1450 472.500 -245255.85 . 64 17 KEY_ARC BASIC 25 2 sales May -1282708.63

Output 4.6.2: DUALOUT=DUAL4

 Obs _node_ _supdem_ _DUAL_ _NNUMB_ _PRED_ _TRAV_ _SCESS_ _ARCID_ _FLOW_ _FBQ_ _VALUE_ _RHS_ _TYPE_ _row_ 1 _ROOT_ 238 0.00 22 0 8 5 3 166.000 -69 0 75 2 _EXCESS_ -200 -100000193.90 21 1 20 13 65 5.000 65 . . 3 f1_apr_1 . -100000278.00 3 1 6 2 4 483.333 4 . . 4 f1_apr_2 . -100000405.92 13 19 11 2 -60 220.000 36 . . 5 f1_mar_1 . -100000326.65 2 8 1 20 -21 143.333 1 . . 6 f1_mar_2 . -100000480.13 12 19 13 1 -59 455.000 33 . . 7 f1_may_1 . -100000284.00 4 1 7 3 8 78.333 8 . . 8 f1_may_2 . -100000349.30 14 17 15 1 -50 350.000 40 . . 9 f2_apr_1 . -100000289.00 6 3 4 1 14 13.333 14 . . 10 f2_apr_2 . -100000415.43 16 20 18 9 47 542.500 46 . . 11 f2_mar_1 . -100000281.90 5 10 3 1 12 255.000 11 . . 12 f2_mar_2 . -100000399.07 15 19 10 1 -62 125.000 43 . . 13 f2_may_1 . -100000300.00 7 4 9 2 18 115.000 18 . . 14 f2_may_2 . -100000375.30 17 19 14 2 -64 472.500 50 . . 15 fact1_1 1000 -100000193.90 1 2 21 19 -1 283.333 -1 . . 16 fact1_2 1000 -100000224.09 11 13 17 1 -36 200.000 -33 . . 17 fact2_1 850 -100000193.90 10 21 5 2 -66 45.000 -33 . . 18 fact2_2 1500 -100000193.90 20 21 16 10 -68 150.000 -65 . . 19 shop1_1 -900 -99999999.00 8 22 2 21 0 0.000 21 . . 20 shop1_2 -900 -99999873.24 18 16 19 1 57 400.000 53 . . 21 shop2_1 -900 -100000001.00 9 7 22 1 32 150.000 27 . . 22 shop2_2 -1450 -99999856.24 19 16 12 7 63 177.500 59 . . 23 . -1.83 2 8 . . 25 243.333 . 2600 2600 LE FACT1 APL GIZMO 24 . -1.62 0 8 . . 23 13.333 . 2600 2600 LE FACT1 MAR GIZMO 25 . -6.21 3 17 . . 51 87.500 . 3750 3750 LE FACT2 APL GIZMO 26 . 0.00 1 1 . 1 . 280.000 . 3470 3750 LE FACT2 MAR GIZMO 27 . -7.42 4 13 . . 38 0.000 . 50 50 LE TOTAL BACKORDER

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