The INTPOINT Procedure

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:

NOTE: The following variables in ARCDATA do not belong to any SAS variable list.

These will be ignored.

_FLOW_

_FCOST_

oldcost

oldfc

oldflow

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 <= side constraints= 5 .

NOTE: Number of == side constraints= 0 .

NOTE: Number of >= side constraints= 0 .

NOTE: Number of side constraint coefficients= 16 .

NOTE: The following messages relate to the equivalent Linear Programming problem

solved by the Interior Point algorithm.

NOTE: Number of <= constraints= 5 .

NOTE: Number of == constraints= 21 .

NOTE: Number of >= constraints= 0 .

NOTE: Number of constraint coefficients= 152 .

NOTE: Number of variables= 68 .

NOTE: After preprocessing, number of <= constraints= 5.

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: 5 columns, 0 rows and 5 coefficients were added to the problem to handle

unrestricted variables, variables that are split, and constraint slack or

surplus variables.

NOTE: There are 74 sub-diagonal nonzeroes in the unfactored A Atranspose matrix.

NOTE: The 25 factor nodes make up 17 supernodes

NOTE: There are 88 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 199456613 0.894741 65408 35351 10906

1 38664128 25735020 0.919726 4738.839318 2561.195456 248.292591

2 5142982 1874540 0.595158 0 0 6.669426

3 366112 338310 0.207256 0 0 1.207816

4 172159 90907 0.063722 0 0 0.238703

5 48403 38889 0.027839 0 0 0.115586

6 28882 17979 0.013029 0 0 0.019825

7 7800.003324 3605.779203 0.002631 0 0 0.004077

8 1564.193112 422.251530 0.000309 0 0 0.000225

9 94.768595 16.589795 0.000012126 0 0 0

10 0.294833 0.001048 5.96523E-10 0 0 0

NOTE: The Primal-Dual Predictor-Corrector Interior Point algorithm performed 10

iterations.

NOTE: Optimum reached.

NOTE: Objective= -1282708.622.

NOTE: The data set WORK.ARC4 has 64 observations and 14 variables.

NOTE: There were 64 observations read from the data set WORK.ARC3.

NOTE: There were 8 observations read from the data set WORK.NODE0.

NOTE: There were 21 observations read from the data set WORK.CON3.

Output 4.3.1: CONOUT=ARC4

Obs	_tail_	_head_	_cost_	_capac_	_lo_	_FLOW_	_FCOST_
1	fact1_1	f1_apr_1	78.60	600	50	533.333	41920.00
2	f1_mar_1	f1_apr_1	15.00	50	0	0.000	0.00
3	f1_may_1	f1_apr_1	33.60	20	0	0.000	0.00
4	f2_apr_1	f1_apr_1	11.00	40	0	0.000	0.00
5	fact1_2	f1_apr_2	174.50	550	50	250.000	43625.00
6	f1_mar_2	f1_apr_2	20.00	40	0	0.000	0.00
7	f1_may_2	f1_apr_2	49.20	15	0	0.000	0.00
8	f2_apr_2	f1_apr_2	21.00	25	0	0.000	0.00
9	fact1_1	f1_mar_1	127.90	500	50	333.333	42633.33
10	f1_apr_1	f1_mar_1	33.60	20	0	20.000	672.00
11	f2_mar_1	f1_mar_1	10.00	40	0	40.000	400.00
12	fact1_2	f1_mar_2	217.90	400	40	400.000	87160.00
13	f1_apr_2	f1_mar_2	38.40	30	0	30.000	1152.00
14	f2_mar_2	f1_mar_2	20.00	25	0	25.000	500.00
15	fact1_1	f1_may_1	90.10	400	50	128.333	11562.83
16	f1_apr_1	f1_may_1	12.00	50	0	0.000	0.00
17	f2_may_1	f1_may_1	13.00	40	0	0.000	0.00
18	fact1_2	f1_may_2	113.30	350	40	350.000	39655.00
19	f1_apr_2	f1_may_2	18.00	40	0	0.000	0.00
20	f2_may_2	f1_may_2	13.00	25	0	0.000	0.00
21	f1_apr_1	f2_apr_1	11.00	99999999	0	13.333	146.67
22	fact2_1	f2_apr_1	62.40	480	35	480.000	29952.00
23	f2_mar_1	f2_apr_1	18.00	30	0	0.000	0.00
24	f2_may_1	f2_apr_1	30.00	15	0	0.000	0.00
25	f1_apr_2	f2_apr_2	23.00	99999999	0	0.000	0.00
26	fact2_2	f2_apr_2	196.70	680	35	577.500	113594.25
27	f2_mar_2	f2_apr_2	28.00	50	0	0.000	0.00
28	f2_may_2	f2_apr_2	64.80	15	0	0.000	0.00
29	f1_mar_1	f2_mar_1	11.00	99999999	0	0.000	0.00
30	fact2_1	f2_mar_1	88.00	450	35	290.000	25520.00
31	f2_apr_1	f2_mar_1	20.40	15	0	0.000	0.00
32	f1_mar_2	f2_mar_2	23.00	99999999	0	0.000	0.00
33	fact2_2	f2_mar_2	182.00	650	35	650.000	118300.00
34	f2_apr_2	f2_mar_2	37.20	15	0	0.000	0.00
35	f1_may_1	f2_may_1	16.00	99999999	0	115.000	1840.00
36	fact2_1	f2_may_1	128.80	250	35	35.000	4508.00
37	f2_apr_1	f2_may_1	20.00	30	0	0.000	0.00
38	f1_may_2	f2_may_2	26.00	99999999	0	350.000	9100.00
39	fact2_2	f2_may_2	181.40	550	35	122.500	22221.50
40	f2_apr_2	f2_may_2	38.00	50	0	0.000	0.00
41	f1_mar_1	shop1_1	-327.65	250	0	143.333	-46963.16
42	f1_apr_1	shop1_1	-300.00	250	0	250.000	-75000.00
43	f1_may_1	shop1_1	-285.00	250	0	13.333	-3800.00
44	f2_mar_1	shop1_1	-297.40	250	0	250.000	-74350.00
45	f2_apr_1	shop1_1	-290.00	250	0	243.333	-70566.67
46	f2_may_1	shop1_1	-292.00	250	0	0.000	0.00
47	f1_mar_2	shop1_2	-559.76	99999999	0	0.000	0.00
48	f1_apr_2	shop1_2	-524.28	99999999	0	0.000	0.00
49	f1_may_2	shop1_2	-475.02	99999999	0	0.000	0.00
50	f2_mar_2	shop1_2	-567.83	500	0	500.000	-283915.00
51	f2_apr_2	shop1_2	-542.19	500	0	400.000	-216876.00
52	f2_may_2	shop1_2	-491.56	500	0	0.000	0.00
53	f1_mar_1	shop2_1	-362.74	250	0	250.000	-90685.00
54	f1_apr_1	shop2_1	-300.00	250	0	250.000	-75000.00
55	f1_may_1	shop2_1	-245.00	250	0	0.000	0.00
56	f2_mar_1	shop2_1	-272.70	250	0	0.000	0.00
57	f2_apr_1	shop2_1	-312.00	250	0	250.000	-78000.00
58	f2_may_1	shop2_1	-299.00	250	0	150.000	-44850.00
59	f1_mar_2	shop2_2	-623.89	99999999	0	455.000	-283869.95
60	f1_apr_2	shop2_2	-549.68	99999999	0	220.000	-120929.60
61	f1_may_2	shop2_2	-460.00	99999999	0	0.000	0.00
62	f2_mar_2	shop2_2	-542.83	500	0	125.000	-67853.75
63	f2_apr_2	shop2_2	-559.19	500	0	177.500	-99256.23
64	f2_may_2	shop2_2	-519.06	500	0	472.500	-245255.85
							-1282708.62