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 The NETFLOW Procedure

## Example 4.8: Nonarc Variables in the Side Constraints

Notice in DUALOUT=dual5 from Example 4.7 the FACT2 MAR GIZMO constraint (observation 24) has a _VALUE_ of 3470, which is not equal to the _RHS_ of this constraint. Not all of the 3750 chips that can be supplied to factory 2 for March production are used. It is suggested that all the possible chips be obtained in March and those not used be saved for April production. Because chips must be kept in an air-controlled environment, it costs 1 dollar to store each chip purchased in March until April. The maximum number of chips that can be stored in this environment at each factory is 150. In addition, a search of the parts inventory at factory 1 turned up 15 chips available for their March production.

Nonarc variables are used in the side constraints that handle the limitations of supply of Gizmo chips. A nonarc variable called "f1 unused mar" has as a value the number of chips that are not used at factory 1 in March. Another nonarc variable, "f2 unused mar", has as a value the number of chips that are not used at factory 2 in March. "f1 chips from mar" has as a value the number of chips left over from March used for production at factory 1 in April. Similarly, "f2 chips from mar" has as a value the number of chips left over from March used for April production at factory 2 in April. The last two nonarc variables have objective function coefficients of 1 and upper bounds of 150. The Gizmo side constraints are

3*prod f1 19 mar + 4*prod f1 25 mar + f1 unused chips = 2615
3*prod f2 19 apl + 4*prod f2 25 apl + f2 unused chips = 3750
3*prod f1 19 apl + 4*prod f1 25 apl - f1 chips from mar = 2600
3*prod f2 19 apl + 4*prod f2 25 apl - f2 chips from mar = 3750
f1 unused chips + f2 unused chips -
f1 chips from mar - f2 chips from mar >= 0

The last side constraint states that the number of chips not used in March is not less than the number of chips left over from March and used in April. Here, this constraint is called CHIP LEFTOVER.

The following SAS code creates a new data set containing constraint data. It seems that most of the constraints are now equalities, so you specify DEFCONTYPE=EQ in the PROC NETFLOW statements from now on and provide constraint type data for constraints that are not "equal to" type, using the default TYPEOBS value _TYPE_ as the _COLUMN_ variable value to indicate observations that contain constraint type data. Also, from now on, the default RHSOBS value is used.

title 'Nonarc Variables in the Side Constraints';
title2 'Production Planning/Inventory/Distribution';
data con6;
input _column_ &\$17. _row_ &\$15. _coef_ ;
datalines;
prod f1 19 mar     FACT1 MAR GIZMO  3
prod f1 25 mar     FACT1 MAR GIZMO  4
f1 unused chips    FACT1 MAR GIZMO  1
_RHS_              FACT1 MAR GIZMO  2615
prod f2 19 mar     FACT2 MAR GIZMO  3
prod f2 25 mar     FACT2 MAR GIZMO  4
f2 unused chips    FACT2 MAR GIZMO  1
_RHS_              FACT2 MAR GIZMO  3750
prod f1 19 apl     FACT1 APL GIZMO  3
prod f1 25 apl     FACT1 APL GIZMO  4
f1 chips from mar  FACT1 APL GIZMO  -1
_RHS_              FACT1 APL GIZMO  2600
prod f2 19 apl     FACT2 APL GIZMO  3
prod f2 25 apl     FACT2 APL GIZMO  4
f2 chips from mar  FACT2 APL GIZMO  -1
_RHS_              FACT2 APL GIZMO  3750
f1 unused chips    CHIP LEFTOVER    1
f2 unused chips    CHIP LEFTOVER    1
f1 chips from mar  CHIP LEFTOVER    -1
f2 chips from mar  CHIP LEFTOVER    -1
_TYPE_             CHIP LEFTOVER    1
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
_TYPE_             TOTAL BACKORDER  -1
_RHS_              TOTAL BACKORDER  50
;

The nonarc variables "f1 chips from mar" and "f2 chips from mar" have objective function coefficients of 1 and upper bounds of 150. There are various ways in which this information can be furnished to PROC NETFLOW. If there were a TYPE list variable in the CONDATA= data set, observations could be in the form:

_COLUMN_           _TYPE_  _ROW_ _COEF_
f1 chips from mar  objfn     .       1
f1 chips from mar  upperbd   .     150
f2 chips from mar  objfn     .       1
f2 chips from mar  upperbd   .     150

It is desireable to assign ID list variable values to all the nonarc variables:

data arc6;
set con5;
drop oldcost oldfc oldflow _flow_ _fcost_ _status_ _rcost_;
data arc6_b;
input _name_ &\$17. _cost_ _capac_ factory key_id \$ ;
datalines;
f1 unused chips    .   . 1 chips
f2 unused chips    .   . 2 chips
f1 chips from mar  1 150 1 chips
f2 chips from mar  1 150 2 chips
;
proc append
base=arc6 data=arc6_b;
proc netflow
nodedata=node0 arcdata=arc6
condata=con6 defcontype=eq sparsecondata
dualout=dual7 conout=con7;
run;
print nonarcs/short;

The following messages appear on the SAS log:

NOTE: Number of nodes= 20 .
NOTE: Number of supply nodes= 4 .
NOTE: Number of demand nodes= 4 .
NOTE: Total supply= 4350 , total demand= 4150 .
NOTE: Number of arcs= 64 .
NOTE: Number of nonarc variables= 8 .
NOTE: Number of iterations performed (neglecting any
constraints)= 70 .
NOTE: Of these, 1 were degenerate.
NOTE: Optimum (neglecting any constraints) found.
NOTE: Minimal total cost= -1295730.8 .
NOTE: Number of <= side constraints= 1 .
NOTE: Number of == side constraints= 4 .
NOTE: Number of >= side constraints= 1 .
NOTE: Number of arc and nonarc variable side constraint
coefficients= 24 .
NOTE: Number of iterations, optimizing with constraints= 13 .
NOTE: Of these, 3 were degenerate.
NOTE: Optimum reached.
NOTE: Minimal total cost= -1295542.742 .
NOTE: The data set WORK.CON7 has 68 observations and 18
variables.
NOTE: The data set WORK.DUAL7 has 26 observations and 14
variables.

The output in Output 4.8.1 is produced by

print nonarcs/short;

Output 4.8.1: Output of PRINT NONARCS/SHORT;

 Nonarc Variables in the Side Constraints Production Planning/Inventory/Distribution

 NETFLOW PROCEDURE

 _N_ _name_ _cost_ _capac_ _lo_ _VALUE_ 1 f1 chips from mar 1 150 0 20 2 f1 unused chips 0 99999999 0 0 3 f2 chips from mar 1 150 0 0 4 f2 unused chips 0 99999999 0 280

The optimal solution data sets, CONOUT=CON7 in Output 4.8.2 and DUALOUT=DUAL7 in Output 4.8.2 follow:

proc print data=con7;
sum _fcost_;
proc print data=dual7;

Output 4.8.2: CONOUT=CON7

 Obs _tail_ _head_ _cost_ _capac_ _lo_ _name_ _SUPPLY_ _DEMAND_ _FLOW_ _FCOST_ _RCOST_ _ANUMB_ _TNUMB_ _STATUS_ diagonal factory key_id mth_made 1 fact1_1 f1_apr_1 78.60 600 50 prod f1 19 apl 1000 . 540.000 42444.00 . 1 1 KEY_ARC BASIC 19 1 production April 2 f1_mar_1 f1_apr_1 15.00 50 0 . . 0.000 0.00 66.150 2 3 LOWERBD NONBASIC 19 1 storage March 3 f1_may_1 f1_apr_1 33.60 20 0 back f1 19 may . . 0.000 0.00 42.580 3 4 LOWERBD NONBASIC 19 1 backorder May 4 f2_apr_1 f1_apr_1 11.00 40 0 . . 0.000 0.00 22.000 4 5 LOWERBD NONBASIC 19 . f2_to_1 April 5 fact1_2 f1_apr_2 174.50 550 50 prod f1 25 apl 1000 . 250.000 43625.00 . 15 6 KEY_ARC BASIC 25 1 production April 6 f1_mar_2 f1_apr_2 20.00 40 0 . . 0.000 0.00 94.210 16 8 LOWERBD NONBASIC 25 1 storage March 7 f1_may_2 f1_apr_2 49.20 15 0 back f1 25 may . . 0.000 0.00 . 17 9 NONKEY ARC BASIC 25 1 backorder May 8 f2_apr_2 f1_apr_2 21.00 25 0 . . 25.000 525.00 -1.510 18 10 UPPERBD NONBASIC 25 . f2_to_1 April 9 fact1_1 f1_mar_1 127.90 500 50 prod f1 19 mar 1000 . 338.333 43272.83 . 5 1 KEY_ARC BASIC 19 1 production March 10 f1_apr_1 f1_mar_1 33.60 20 0 back f1 19 apl . . 20.000 672.00 -17.070 6 2 UPPERBD NONBASIC 19 1 backorder April 11 f2_mar_1 f1_mar_1 10.00 40 0 . . 40.000 400.00 -34.750 7 11 UPPERBD NONBASIC 19 . f2_to_1 March 12 fact1_2 f1_mar_2 217.90 400 40 prod f1 25 mar 1000 . 400.000 87160.00 -28.343 19 6 UPPERBD NONBASIC 25 1 production March 13 f1_apr_2 f1_mar_2 38.40 30 0 back f1 25 apl . . 30.000 1152.00 -35.330 20 7 UPPERBD NONBASIC 25 1 backorder April 14 f2_mar_2 f1_mar_2 20.00 25 0 . . 25.000 500.00 -61.060 21 12 UPPERBD NONBASIC 25 . f2_to_1 March 15 fact1_1 f1_may_1 90.10 400 50 1000 . 116.667 10511.67 . 8 1 KEY_ARC BASIC 19 1 production May 16 f1_apr_1 f1_may_1 12.00 50 0 . . 0.000 0.00 3.500 9 2 LOWERBD NONBASIC 19 1 storage April 17 f2_may_1 f1_may_1 13.00 40 0 . . 0.000 0.00 29.000 10 13 LOWERBD NONBASIC 19 . f2_to_1 May 18 fact1_2 f1_may_2 113.30 350 40 1000 . 350.000 39655.00 -15.520 22 6 UPPERBD NONBASIC 25 1 production May 19 f1_apr_2 f1_may_2 18.00 40 0 . . 0.000 0.00 67.680 23 7 LOWERBD NONBASIC 25 1 storage April 20 f2_may_2 f1_may_2 13.00 25 0 . . 0.000 0.00 32.060 24 14 LOWERBD NONBASIC 25 . f2_to_1 May 21 f1_apr_1 f2_apr_1 11.00 99999999 0 . . 20.000 220.00 . 11 2 KEY_ARC BASIC 19 . f1_to_2 April 22 fact2_1 f2_apr_1 62.40 480 35 prod f2 19 apl 850 . 480.000 29952.00 -35.592 12 15 UPPERBD NONBASIC 19 2 production April 23 f2_mar_1 f2_apr_1 18.00 30 0 . . 0.000 0.00 13.400 13 11 LOWERBD NONBASIC 19 2 storage March 24 f2_may_1 f2_apr_1 30.00 15 0 back f2 19 may . . 0.000 0.00 43.980 14 13 LOWERBD NONBASIC 19 2 backorder May 25 f1_apr_2 f2_apr_2 23.00 99999999 0 . . 0.000 0.00 45.510 25 7 LOWERBD NONBASIC 25 . f1_to_2 April 26 fact2_2 f2_apr_2 196.70 680 35 prod f2 25 apl 1500 . 577.500 113594.25 . 26 16 KEY_ARC BASIC 25 2 production April 27 f2_mar_2 f2_apr_2 28.00 50 0 . . 0.000 0.00 43.660 27 12 LOWERBD NONBASIC 25 2 storage March 28 f2_may_2 f2_apr_2 64.80 15 0 back f2 25 may . . 0.000 0.00 57.170 28 14 LOWERBD NONBASIC 25 2 backorder May 29 f1_mar_1 f2_mar_1 11.00 99999999 0 . . 0.000 0.00 55.750 29 3 LOWERBD NONBASIC 19 . f1_to_2 March 30 fact2_1 f2_mar_1 88.00 450 35 prod f2 19 mar 850 . 290.000 25520.00 . 30 15 KEY_ARC BASIC 19 2 production March 31 f2_apr_1 f2_mar_1 20.40 15 0 back f2 19 apl . . 0.000 0.00 25.480 31 5 LOWERBD NONBASIC 19 2 backorder April 32 f1_mar_2 f2_mar_2 23.00 99999999 0 . . 0.000 0.00 104.060 32 8 LOWERBD NONBASIC 25 . f1_to_2 March 33 fact2_2 f2_mar_2 182.00 650 35 prod f2 25 mar 1500 . 650.000 118300.00 -23.170 33 16 UPPERBD NONBASIC 25 2 production March 34 f2_apr_2 f2_mar_2 37.20 15 0 back f2 25 apl . . 0.000 0.00 22.020 34 10 LOWERBD NONBASIC 25 2 backorder April 35 f1_may_1 f2_may_1 16.00 99999999 0 . . 115.000 1840.00 . 35 4 KEY_ARC BASIC 19 . f1_to_2 May 36 fact2_1 f2_may_1 128.80 250 35 850 . 35.000 4508.00 22.700 36 15 LOWERBD NONBASIC 19 2 production May 37 f2_apr_1 f2_may_1 20.00 30 0 . . 0.000 0.00 6.500 37 5 LOWERBD NONBASIC 19 2 storage April 38 f1_may_2 f2_may_2 26.00 99999999 0 . . 0.000 0.00 6.940 38 9 LOWERBD NONBASIC 25 . f1_to_2 May 39 fact2_2 f2_may_2 181.40 550 35 1500 . 122.500 22221.50 . 39 16 KEY_ARC BASIC 25 2 production May 40 f2_apr_2 f2_may_2 38.00 50 0 . . 0.000 0.00 46.110 40 10 LOWERBD NONBASIC 25 2 storage April 41 f1_mar_1 shop1_1 -327.65 250 0 . 900 148.333 -48601.42 . 41 3 KEY_ARC BASIC 19 1 sales March 42 f1_apr_1 shop1_1 -300.00 250 0 . 900 250.000 -75000.00 -23.500 42 2 UPPERBD NONBASIC 19 1 sales April 43 f1_may_1 shop1_1 -285.00 250 0 . 900 1.667 -475.00 . 43 4 NONKEY ARC BASIC 19 1 sales May 44 f2_mar_1 shop1_1 -297.40 250 0 . 900 250.000 -74350.00 -14.500 44 11 UPPERBD NONBASIC 19 2 sales March 45 f2_apr_1 shop1_1 -290.00 250 0 . 900 250.000 -72500.00 -2.500 45 5 UPPERBD NONBASIC 19 2 sales April 46 f2_may_1 shop1_1 -292.00 250 0 . 900 0.000 0.00 9.000 46 13 LOWERBD NONBASIC 19 2 sales May 47 f1_mar_2 shop1_2 -559.76 99999999 0 . 900 0.000 0.00 79.150 47 8 LOWERBD NONBASIC 25 1 sales March 48 f1_apr_2 shop1_2 -524.28 99999999 0 . 900 0.000 0.00 40.420 48 7 LOWERBD NONBASIC 25 1 sales April 49 f1_may_2 shop1_2 -515.02 99999999 0 . 900 347.500 -178969.45 . 49 9 KEY_ARC BASIC 25 1 sales May 50 f2_mar_2 shop1_2 -567.83 500 0 . 900 500.000 -283915.00 -9.980 50 12 UPPERBD NONBASIC 25 2 sales March 51 f2_apr_2 shop1_2 -542.19 500 0 . 900 52.500 -28464.98 . 51 10 KEY_ARC BASIC 25 2 sales April 52 f2_may_2 shop1_2 -491.56 500 0 . 900 0.000 0.00 42.520 52 14 LOWERBD NONBASIC 25 2 sales May 53 f1_mar_1 shop2_1 -362.74 250 0 . 900 250.000 -90685.00 -37.090 53 3 UPPERBD NONBASIC 19 1 sales March 54 f1_apr_1 shop2_1 -300.00 250 0 . 900 250.000 -75000.00 -25.500 54 2 UPPERBD NONBASIC 19 1 sales April 55 f1_may_1 shop2_1 -245.00 250 0 . 900 0.000 0.00 38.000 55 4 LOWERBD NONBASIC 19 1 sales May 56 f2_mar_1 shop2_1 -272.70 250 0 . 900 0.000 0.00 8.200 56 11 LOWERBD NONBASIC 19 2 sales March 57 f2_apr_1 shop2_1 -312.00 250 0 . 900 250.000 -78000.00 -26.500 57 5 UPPERBD NONBASIC 19 2 sales April 58 f2_may_1 shop2_1 -299.00 250 0 . 900 150.000 -44850.00 . 58 13 KEY_ARC BASIC 19 2 sales May 59 f1_mar_2 shop2_2 -623.89 99999999 0 . 1450 455.000 -283869.95 . 59 8 KEY_ARC BASIC 25 1 sales March 60 f1_apr_2 shop2_2 -549.68 99999999 0 . 1450 245.000 -134671.60 . 60 7 KEY_ARC BASIC 25 1 sales April 61 f1_may_2 shop2_2 -500.00 99999999 0 . 1450 2.500 -1250.00 . 61 9 NONKEY ARC BASIC 25 1 sales May 62 f2_mar_2 shop2_2 -542.83 500 0 . 1450 125.000 -67853.75 . 62 12 KEY_ARC BASIC 25 2 sales March 63 f2_apr_2 shop2_2 -559.19 500 0 . 1450 500.000 -279595.00 -32.020 63 10 UPPERBD NONBASIC 25 2 sales April 64 f2_may_2 shop2_2 -519.06 500 0 . 1450 122.500 -63584.85 . 64 14 KEY_ARC BASIC 25 2 sales May 65 1.00 150 0 f1 chips from mar . . 20.000 20.00 . -2 . NONKEY BASIC . 1 chips 66 0.00 99999999 0 f1 unused chips . . 0.000 0.00 1.617 0 . LOWERBD NONBASIC . 1 chips 67 1.00 150 0 f2 chips from mar . . 0.000 0.00 2.797 -3 . LOWERBD NONBASIC . 2 chips 68 0.00 99999999 0 f2 unused chips . . 280.000 0.00 . -1 . NONKEY BASIC . 2 chips -1295542.74

Output 4.8.3: DUALOUT=DUAL7

 Obs _node_ _supdem_ _DUAL_ _NNUMB_ _PRED_ _TRAV_ _SCESS_ _ARCID_ _FLOW_ _FBQ_ _VALUE_ _RHS_ _TYPE_ _row_ 1 f1_apr_1 . -100000275.50 2 1 5 2 1 490.000 1 . . 2 f1_apr_2 . -100000405.92 7 20 6 2 -60 245.000 15 . . 3 f1_mar_1 . -100000326.65 3 17 1 20 -41 148.333 5 . . 4 f1_mar_2 . -100000480.13 8 20 7 1 -59 455.000 19 . . 5 f1_may_1 . -100000284.00 4 1 13 3 8 66.667 8 . . 6 f1_may_2 . -100000356.24 9 18 2 1 -49 347.500 22 . . 7 f2_apr_1 . -100000286.50 5 2 4 1 11 20.000 11 . . 8 f2_apr_2 . -100000383.41 10 16 18 3 26 542.500 25 . . 9 f2_mar_1 . -100000281.90 11 15 16 1 30 255.000 29 . . 10 f2_mar_2 . -100000399.07 12 20 10 1 -62 125.000 32 . . 11 f2_may_1 . -100000300.00 13 4 19 2 35 115.000 35 . . 12 f2_may_2 . -100000375.30 14 16 20 6 39 87.500 38 . . 13 fact1_1 1000 -100000193.90 1 3 21 19 -5 288.333 -1 . . 14 fact1_2 1000 -100000227.42 6 7 12 1 -15 200.000 -15 . . 15 fact2_1 850 -100000193.90 15 21 11 2 -67 45.000 -41 . . 16 fact2_2 1500 -100000193.90 16 21 14 10 -68 150.000 -41 . . 17 shop1_1 -900 -99999999.00 17 22 3 21 0 0.000 41 . . 18 shop1_2 -900 -99999841.22 18 10 9 2 51 52.500 47 . . 19 shop2_1 -900 -100000001.00 19 13 22 1 58 150.000 53 . . 20 shop2_2 -1450 -99999856.24 20 14 8 5 64 122.500 59 . . 21 . 0.00 4 4 . 4 . 260.000 . 260 0 GE CHIP LEFTOVER 22 . -1.00 2 2 . . -2 20.000 . 2600 2600 EQ FACT1 APL GIZMO 23 . -1.62 0 17 . . 43 1.667 . 2615 2615 EQ FACT1 MAR GIZMO 24 . 1.80 3 20 . . 61 2.500 . 3750 3750 EQ FACT2 APL GIZMO 25 . 0.00 1 1 . . -1 280.000 . 3750 3750 EQ FACT2 MAR GIZMO 26 . -0.48 5 7 . . 17 0.000 . 50 50 LE TOTAL BACKORDER

The optimal value of the nonarc variable "f2 unused chips" is 280. This means that although there are 3750 chips that can be used at factory 2 in March, only 3470 are used. As the optimal value of "f1 unused chips" is zero, all chips available for production in March at factory 1 are used. The nonarc variable "f2 chips from mar" also has zero optimal value. This means that the April production at factory 2 does not need any chips that could have been held in inventory since March. However, the nonarc variable "f1 chips from mar" has value of 20. Thus, 3490 chips should be ordered for factory 2 in March. Twenty of these chips should be held in inventory until April, then sent to factory 1.

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