### Example 5.2 Altering Arc Data

This example examines the effect of changing some of the arc costs. The backorder penalty costs are increased by 20 percent. The sales profit of 25-inch TVs sent to the shops in May is increased by 30 units. The backorder penalty costs of 25-inch TVs manufactured in May for April consumption is decreased by 30 units. The production cost of 19-inch and 25-inch TVs made in May are decreased by 5 units and 20 units, respectively. How does the optimal solution of the network after these arc cost alterations compare with the optimum of the original network?

These SAS statements produce the new NODEDATA= and ARCDATA= data sets:

```title2 'Minimum Cost Flow problem- Altered Arc Data';
data arc2;
set arc1;
oldcost=_cost_;
oldfc=_fcost_;
oldflow=_flow_;
if key_id='backorder'
then _cost_=_cost_*1.2;
else if _tail_='f2_may_2' then _cost_=_cost_-30;
if diagonal=19 then _cost_=_cost_-5;
else _cost_=_cost_-20;
run;
```
```proc intpoint
bytes=100000
printlevel2=2
nodedata=node0
arcdata=arc2
conout=arc3;
run;
```
```proc print data=arc3;
var _tail_ _head_ _capac_ _lo_ _supply_ _demand_ _name_
_cost_ _flow_ _fcost_ oldcost oldflow oldfc
/* to get this variable order */
sum oldfc _fcost_;
run;
```

The following notes appear on the SAS log:

 Production Planning/Inventory/Distribution Minimum Cost Flow problem- Altered Arc Data

 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: The following messages relate to the equivalent Linear Programming problem solved by the Interior Point algorithm. NOTE: Number of <= constraints= 0 . NOTE: Number of == constraints= 21 . NOTE: Number of >= constraints= 0 . NOTE: Number of constraint coefficients= 136 . NOTE: Number of variables= 68 . NOTE: After preprocessing, number of <= constraints= 0. NOTE: After preprocessing, number of == constraints= 20. NOTE: After preprocessing, number of >= constraints= 0. NOTE: The preprocessor eliminated 1 constraints from the problem. NOTE: The preprocessor eliminated 9 constraint coefficients from the problem. NOTE: 0 columns, 0 rows and 0 coefficients were added to the problem to handle unrestricted variables, variables that are split, and constraint slack or surplus variables. NOTE: There are 48 sub-diagonal nonzeroes in the unfactored A Atranspose matrix. NOTE: The 20 factor nodes make up 8 supernodes NOTE: There are 27 nonzero sub-rows or sub-columns outside the supernodal triangular regions along the factors leading diagonal. Iter  Complem_aff  Complem-ity  Duality_gap  Tot_infeasb  Tot_infeasc  Tot_infeasd 0    -1.000000    193775969     0.894415        66024        25664            0 1     37797544     24594220     0.918149  4566.893212  1775.179450            0 2      4408681      1844606     0.590964            0            0            0 3       347168       312126     0.194113            0            0            0 4       145523        86002     0.060330            0            0            0 5        43008        38240     0.027353            0            0            0 6        31097        21145     0.015282            0            0            0 7  9308.807034  4158.399675     0.003029            0            0            0 8  1710.832075   752.174595     0.000549            0            0            0 9   254.197112    47.755299  0.000034846            0            0            0 10     5.252560     0.010692 7.8017573E-9            0            0            0 NOTE: The Primal-Dual Predictor-Corrector Interior Point algorithm performed 10 iterations. NOTE: Optimum reached. NOTE: Objective= -1285086.442. NOTE: The data set WORK.ARC3 has 64 observations and 17 variables. NOTE: There were 64 observations read from the data set WORK.ARC2. NOTE: There were 8 observations read from the data set WORK.NODE0.

The solution is displayed in Output 5.2.1.

Output 5.2.1: CONOUT=ARC3

 Production Planning/Inventory/Distribution Minimum Cost Flow problem- Altered arc data

_tail_ _head_ _capac_ _lo_ _SUPPLY_ _DEMAND_ _name_ _cost_ _FLOW_
fact1_1 f1_apr_1 600 50 1000 . prod f1 19 apl 78.60 540.000
f1_mar_1 f1_apr_1 50 0 . .   15.00 0.000
f1_may_1 f1_apr_1 20 0 . . back f1 19 may 33.60 0.000
f2_apr_1 f1_apr_1 40 0 . .   11.00 0.000
fact1_2 f1_apr_2 550 50 1000 . prod f1 25 apl 174.50 250.000
f1_mar_2 f1_apr_2 40 0 . .   20.00 0.000
f1_may_2 f1_apr_2 15 0 . . back f1 25 may 49.20 15.000
f2_apr_2 f1_apr_2 25 0 . .   21.00 0.000
fact1_1 f1_mar_1 500 50 1000 . prod f1 19 mar 127.90 340.000
f1_apr_1 f1_mar_1 20 0 . . back f1 19 apl 33.60 20.000
f2_mar_1 f1_mar_1 40 0 . .   10.00 40.000
fact1_2 f1_mar_2 400 40 1000 . prod f1 25 mar 217.90 400.000
f1_apr_2 f1_mar_2 30 0 . . back f1 25 apl 38.40 30.000
f2_mar_2 f1_mar_2 25 0 . .   20.00 25.000
fact1_1 f1_may_1 400 50 1000 .   90.10 115.000
f1_apr_1 f1_may_1 50 0 . .   12.00 0.000
f2_may_1 f1_may_1 40 0 . .   13.00 0.000
fact1_2 f1_may_2 350 40 1000 .   113.30 350.000
f1_apr_2 f1_may_2 40 0 . .   18.00 0.000
f2_may_2 f1_may_2 25 0 . .   13.00 0.000
f1_apr_1 f2_apr_1 99999999 0 . .   11.00 20.000
fact2_1 f2_apr_1 480 35 850 . prod f2 19 apl 62.40 480.000
f2_mar_1 f2_apr_1 30 0 . .   18.00 0.000
f2_may_1 f2_apr_1 15 0 . . back f2 19 may 30.00 0.000
f1_apr_2 f2_apr_2 99999999 0 . .   23.00 0.000
fact2_2 f2_apr_2 680 35 1500 . prod f2 25 apl 196.70 680.000
f2_mar_2 f2_apr_2 50 0 . .   28.00 0.000
f2_may_2 f2_apr_2 15 0 . . back f2 25 may 64.80 0.000
f1_mar_1 f2_mar_1 99999999 0 . .   11.00 0.000
fact2_1 f2_mar_1 450 35 850 . prod f2 19 mar 88.00 290.000
f2_apr_1 f2_mar_1 15 0 . . back f2 19 apl 20.40 0.000
f1_mar_2 f2_mar_2 99999999 0 . .   23.00 0.000
fact2_2 f2_mar_2 650 35 1500 . prod f2 25 mar 182.00 635.000
f2_apr_2 f2_mar_2 15 0 . . back f2 25 apl 37.20 0.000
f1_may_1 f2_may_1 99999999 0 . .   16.00 115.000
fact2_1 f2_may_1 250 35 850 .   128.80 35.000
f2_apr_1 f2_may_1 30 0 . .   20.00 0.000
f1_may_2 f2_may_2 99999999 0 . .   26.00 335.000
fact2_2 f2_may_2 550 35 1500 .   181.40 35.000
f2_apr_2 f2_may_2 50 0 . .   38.00 0.000
f1_mar_1 shop1_1 250 0 . 900   -327.65 150.000
f1_apr_1 shop1_1 250 0 . 900   -300.00 250.000
f1_may_1 shop1_1 250 0 . 900   -285.00 0.000
f2_mar_1 shop1_1 250 0 . 900   -297.40 250.000
f2_apr_1 shop1_1 250 0 . 900   -290.00 250.000
f2_may_1 shop1_1 250 0 . 900   -292.00 0.000
f1_mar_2 shop1_2 99999999 0 . 900   -559.76 0.000
f1_apr_2 shop1_2 99999999 0 . 900   -524.28 0.000
f1_may_2 shop1_2 99999999 0 . 900   -475.02 0.000
f2_mar_2 shop1_2 500 0 . 900   -567.83 500.000
f2_apr_2 shop1_2 500 0 . 900   -542.19 400.000
f2_may_2 shop1_2 500 0 . 900   -491.56 0.000
f1_mar_1 shop2_1 250 0 . 900   -362.74 250.000
f1_apr_1 shop2_1 250 0 . 900   -300.00 250.000
f1_may_1 shop2_1 250 0 . 900   -245.00 0.000
f2_mar_1 shop2_1 250 0 . 900   -272.70 0.000
f2_apr_1 shop2_1 250 0 . 900   -312.00 250.000
f2_may_1 shop2_1 250 0 . 900   -299.00 150.000
f1_mar_2 shop2_2 99999999 0 . 1450   -623.89 455.000
f1_apr_2 shop2_2 99999999 0 . 1450   -549.68 235.000
f1_may_2 shop2_2 99999999 0 . 1450   -460.00 0.000
f2_mar_2 shop2_2 500 0 . 1450   -542.83 110.000
f2_apr_2 shop2_2 500 0 . 1450   -559.19 280.000
f2_may_2 shop2_2 500 0 . 1450   -519.06 370.000

 Production Planning/Inventory/Distribution Minimum Cost Flow problem- Altered arc data

Obs _FCOST_ oldcost oldflow oldfc diagonal factory key_id mth_made
1 42444.01 78.60 600.000 47160.00 19 1 production April
2 0.00 15.00 0.000 0.00 19 1 storage March
3 0.00 28.00 0.000 0.00 19 1 backorder May
4 0.00 11.00 0.000 0.00 19 . f2_to_1 April
5 43625.00 174.50 550.000 95975.00 25 1 production April
6 0.00 20.00 0.000 0.00 25 1 storage March
7 738.00 41.00 15.000 615.00 25 1 backorder May
8 0.00 21.00 0.000 0.00 25 . f2_to_1 April
9 43486.02 127.90 344.999 44125.43 19 1 production March
10 672.00 28.00 20.000 560.00 19 1 backorder April
11 400.00 10.00 40.000 400.00 19 . f2_to_1 March
12 87160.00 217.90 400.000 87160.00 25 1 production March
13 1152.00 32.00 30.000 960.00 25 1 backorder April
14 500.00 20.00 25.000 500.00 25 . f2_to_1 March
15 10361.47 95.10 50.001 4755.06 19 1 production May
16 0.00 12.00 50.000 600.00 19 1 storage April
17 0.00 13.00 0.000 0.00 19 . f2_to_1 May
18 39655.00 133.30 40.000 5332.04 25 1 production May
19 0.00 18.00 0.000 0.00 25 1 storage April
20 0.00 43.00 0.000 0.00 25 . f2_to_1 May
21 220.00 11.00 30.000 330.00 19 . f1_to_2 April
22 29952.00 62.40 480.000 29952.00 19 2 production April
23 0.00 18.00 0.000 0.00 19 2 storage March
24 0.00 25.00 0.000 0.00 19 2 backorder May
25 0.00 23.00 0.000 0.00 25 . f1_to_2 April
26 133755.99 196.70 680.000 133755.99 25 2 production April
27 0.00 28.00 0.000 0.00 25 2 storage March
28 0.00 54.00 15.000 810.00 25 2 backorder May
29 0.00 11.00 0.000 0.00 19 . f1_to_2 March
30 25520.00 88.00 290.000 25520.00 19 2 production March
31 0.00 17.00 0.000 0.00 19 2 backorder April
32 0.00 23.00 0.000 0.00 25 . f1_to_2 March
33 115570.01 182.00 645.000 117389.96 25 2 production March
34 0.00 31.00 0.000 0.00 25 2 backorder April
35 1840.00 16.00 100.000 1600.01 19 . f1_to_2 May
36 4508.00 133.80 35.000 4683.00 19 2 production May
37 0.00 20.00 15.000 299.99 19 2 storage April
38 8710.00 26.00 0.000 0.00 25 . f1_to_2 May
39 6349.00 201.40 35.000 7049.00 25 2 production May
40 0.00 38.00 0.000 0.00 25 2 storage April
41 -49147.54 -327.65 154.999 -50785.56 19 1 sales March
42 -75000.00 -300.00 250.000 -75000.00 19 1 sales April
43 -0.01 -285.00 0.000 0.00 19 1 sales May
44 -74350.00 -297.40 250.000 -74349.99 19 2 sales March
45 -72499.96 -290.00 245.001 -71050.17 19 2 sales April
46 0.00 -292.00 0.000 0.00 19 2 sales May
47 0.00 -559.76 0.000 0.00 25 1 sales March
48 -0.01 -524.28 0.000 -0.01 25 1 sales April
49 -0.06 -475.02 25.000 -11875.64 25 1 sales May
50 -283915.00 -567.83 500.000 -283915.00 25 2 sales March
51 -216875.92 -542.19 375.000 -203321.08 25 2 sales April
52 0.00 -461.56 0.000 0.00 25 2 sales May
53 -90685.00 -362.74 250.000 -90685.00 19 1 sales March
54 -75000.00 -300.00 250.000 -75000.00 19 1 sales April
55 0.00 -245.00 0.000 0.00 19 1 sales May
56 -0.01 -272.70 0.000 0.00 19 2 sales March
57 -78000.00 -312.00 250.000 -78000.00 19 2 sales April
58 -44849.99 -299.00 150.000 -44850.00 19 2 sales May
59 -283869.94 -623.89 455.000 -283869.94 25 1 sales March
60 -129174.80 -549.68 535.000 -294078.78 25 1 sales April
61 0.00 -460.00 0.000 0.00 25 1 sales May
62 -59711.32 -542.83 120.000 -65139.47 25 2 sales March
63 -156573.27 -559.19 320.000 -178940.96 25 2 sales April
64 -192052.13 -489.06 20.000 -9781.20 25 2 sales May
-1285086.44     -1281110.34