Example 5.5 Nonarc Variables in the Side Constraints :: SAS/OR(R) 12.1 User's Guide: Mathematical Programming Legacy Procedures

Example 5.5 Nonarc Variables in the Side Constraints

You can verify that the FACT2 MAR GIZMO constraint has a left-hand-side activity of 3,470, which is not equal to the _RHS_ of this constraint. Not all of the 3,750 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 one 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 chips has as a value the number of chips that are not used at factory 1 in March. Another nonarc variable, f2 unused chips, 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 INTPOINT statement 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.

title2 'Nonarc Variables in the Side Constraints';
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 INTPOINT. 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 desirable to assign ID list variable values to all the nonarc variables:

data arc6;
   input _tail_ $ _head_ $ _cost_ _capac_ _lo_  diagonal factory
         key_id $10. mth_made $ _name_&$17.;
datalines;
fact1_1  f1_apr_1   78.60 600 50 19 1 production April prod f1 19 apl
f1_mar_1 f1_apr_1   15.00  50  . 19 1 storage    March .
f1_may_1 f1_apr_1   33.60  20  . 19 1 backorder  May   back f1 19 may
f2_apr_1 f1_apr_1   11.00  40  . 19 . f2_to_1    April .
fact1_2  f1_apr_2  174.50 550 50 25 1 production April prod f1 25 apl
f1_mar_2 f1_apr_2   20.00  40  . 25 1 storage    March .
f1_may_2 f1_apr_2   49.20  15  . 25 1 backorder  May   back f1 25 may
f2_apr_2 f1_apr_2   21.00  25  . 25 . f2_to_1    April .
fact1_1  f1_mar_1  127.90 500 50 19 1 production March prod f1 19 mar
f1_apr_1 f1_mar_1   33.60  20  . 19 1 backorder  April back f1 19 apl
f2_mar_1 f1_mar_1   10.00  40  . 19 . f2_to_1    March .
fact1_2  f1_mar_2  217.90 400 40 25 1 production March prod f1 25 mar
f1_apr_2 f1_mar_2   38.40  30  . 25 1 backorder  April back f1 25 apl
f2_mar_2 f1_mar_2   20.00  25  . 25 . f2_to_1    March .
fact1_1  f1_may_1   90.10 400 50 19 1 production May   .
f1_apr_1 f1_may_1   12.00  50  . 19 1 storage    April .
f2_may_1 f1_may_1   13.00  40  . 19 . f2_to_1    May   .
fact1_2  f1_may_2  113.30 350 40 25 1 production May   .
f1_apr_2 f1_may_2   18.00  40  . 25 1 storage    April .
f2_may_2 f1_may_2   13.00  25  . 25 . f2_to_1    May   .
f1_apr_1 f2_apr_1   11.00   .  . 19 . f1_to_2    April .
fact2_1  f2_apr_1   62.40 480 35 19 2 production April prod f2 19 apl
f2_mar_1 f2_apr_1   18.00  30  . 19 2 storage    March .
f2_may_1 f2_apr_1   30.00  15  . 19 2 backorder  May   back f2 19 may
f1_apr_2 f2_apr_2   23.00   .  . 25 . f1_to_2    April .
fact2_2  f2_apr_2  196.70 680 35 25 2 production April prod f2 25 apl
f2_mar_2 f2_apr_2   28.00  50  . 25 2 storage    March .
f2_may_2 f2_apr_2   64.80  15  . 25 2 backorder  May   back f2 25 may
f1_mar_1 f2_mar_1   11.00   .  . 19 . f1_to_2    March .
fact2_1  f2_mar_1   88.00 450 35 19 2 production March prod f2 19 mar
f2_apr_1 f2_mar_1   20.40  15  . 19 2 backorder  April back f2 19 apl
f1_mar_2 f2_mar_2   23.00   .  . 25 . f1_to_2    March .
fact2_2  f2_mar_2  182.00 650 35 25 2 production March prod f2 25 mar
f2_apr_2 f2_mar_2   37.20  15  . 25 2 backorder  April back f2 25 apl
f1_may_1 f2_may_1   16.00   .  . 19 . f1_to_2    May   .
fact2_1  f2_may_1  128.80 250 35 19 2 production May   .
f2_apr_1 f2_may_1   20.00  30  . 19 2 storage    April .
f1_may_2 f2_may_2   26.00   .  . 25 . f1_to_2    May   .
fact2_2  f2_may_2  181.40 550 35 25 2 production May   .
f2_apr_2 f2_may_2   38.00  50  . 25 2 storage    April .
f1_mar_1 shop1_1  -327.65 250  . 19 1 sales      March .
f1_apr_1 shop1_1  -300.00 250  . 19 1 sales      April .
f1_may_1 shop1_1  -285.00 250  . 19 1 sales      May   .
f2_mar_1 shop1_1  -297.40 250  . 19 2 sales      March .
f2_apr_1 shop1_1  -290.00 250  . 19 2 sales      April .
f2_may_1 shop1_1  -292.00 250  . 19 2 sales      May   .
f1_mar_2 shop1_2  -559.76   .  . 25 1 sales      March .
f1_apr_2 shop1_2  -524.28   .  . 25 1 sales      April .
f1_may_2 shop1_2  -515.02   .  . 25 1 sales      May   .
f2_mar_2 shop1_2  -567.83 500  . 25 2 sales      March .
f2_apr_2 shop1_2  -542.19 500  . 25 2 sales      April .
f2_may_2 shop1_2  -491.56 500  . 25 2 sales      May   .
f1_mar_1 shop2_1  -362.74 250  . 19 1 sales      March .
f1_apr_1 shop2_1  -300.00 250  . 19 1 sales      April .
f1_may_1 shop2_1  -245.00 250  . 19 1 sales      May   .
f2_mar_1 shop2_1  -272.70 250  . 19 2 sales      March .
f2_apr_1 shop2_1  -312.00 250  . 19 2 sales      April .
f2_may_1 shop2_1  -299.00 250  . 19 2 sales      May   .
f1_mar_2 shop2_2  -623.89   .  . 25 1 sales      March .
f1_apr_2 shop2_2  -549.68   .  . 25 1 sales      April .
f1_may_2 shop2_2  -500.00   .  . 25 1 sales      May   .
f2_mar_2 shop2_2  -542.83 500  . 25 2 sales      March .
f2_apr_2 shop2_2  -559.19 500  . 25 2 sales      April .
f2_may_2 shop2_2  -519.06 500  . 25 2 sales      May   .
;

data arc6;
   set arc5;
   drop oldcost oldfc oldflow _flow_ _fcost_ ;
   run;

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 force
   base=arc6 data=arc6_b;
   run;
proc intpoint
   bytes=1000000 
   printlevel2=2
   nodedata=node0 arcdata=arc6
   condata=con6 defcontype=eq sparsecondata
   conout=arc7;
   run;

The following messages appear on the SAS log:

Production Planning/Inventory/Distribution

Nonarc Variables in the Side Constraints

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= 4 .

NOTE: Number of <= side constraints= 1 .

NOTE: Number of == side constraints= 4 .

NOTE: Number of >= side constraints= 1 .

NOTE: Number of side constraint coefficients= 24 .

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

solved by the Interior Point algorithm.

NOTE: Number of <= constraints= 1 .

NOTE: Number of == constraints= 25 .

NOTE: Number of >= constraints= 1 .

NOTE: Number of constraint coefficients= 160 .

NOTE: Number of variables= 72 .

NOTE: After preprocessing, number of <= constraints= 1.

NOTE: After preprocessing, number of == constraints= 24.

NOTE: After preprocessing, number of >= constraints= 1.

NOTE: The preprocessor eliminated 1 constraints from the problem.

NOTE: The preprocessor eliminated 9 constraint coefficients from the problem.

NOTE: 2 columns, 0 rows and 2 coefficients were added to the problem to handle

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

surplus variables.

NOTE: There are 78 sub-diagonal nonzeroes in the unfactored A Atranspose matrix.

NOTE: The 26 factor nodes make up 18 supernodes

NOTE: There are 101 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 210688061 0.904882 69336 35199 4398.024971

1 54066756 35459986 0.931873 5967.706945 3029.541352 935.225890

2 10266927 2957978 0.671565 0 0 36.655485

3 326659 314818 0.177750 0 0 3.893178

4 137432 83570 0.053111 0 0 0.852994

5 41386 26985 0.017545 0 0 0.204166

6 12451 6063.528974 0.003973 0 0 0.041229

7 2962.309960 1429.369437 0.000939 0 0 0.004395

8 352.469864 233.620884 0.000153 0 0 0.000297

9 115.012309 23.329492 0.000015331 0 0 0

10 1.754859 0.039304 2.5828276E-8 0 0 0

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

iterations.

NOTE: Optimum reached.

NOTE: Objective= -1295542.717.

NOTE: The data set WORK.ARC7 has 68 observations and 14 variables.

NOTE: There were 68 observations read from the data set WORK.ARC6.

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

NOTE: There were 31 observations read from the data set WORK.CON6.

The optimal solution data set, CONOUT=ARC7, is given in Output 5.5.1.

proc print data=arc7;
   var _tail_ _head_ _name_ _cost_ _capac_ _lo_
       _flow_ _fcost_;
   sum _fcost_;
   run;

The optimal value of the nonarc variable f2 unused chips is 280. This means that although there are 3,750 chips that can be used at factory 2 in March, only 3,470 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, 3,490 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.

Output 5.5.1: CONOUT=ARC7

Production Planning/Inventory/Distribution

Nonarc Variables in the Side Constraints

Obs	_tail_	_head_	_name_	_cost_	_capac_	_lo_	_FLOW_	_FCOST_
1	fact1_1	f1_apr_1	prod f1 19 apl	78.60	600	50	540.000	42444.00
2	f1_mar_1	f1_apr_1		15.00	50	0	0.000	0.00
3	f1_may_1	f1_apr_1	back f1 19 may	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	prod f1 25 apl	174.50	550	50	250.000	43625.01
6	f1_mar_2	f1_apr_2		20.00	40	0	0.000	0.00
7	f1_may_2	f1_apr_2	back f1 25 may	49.20	15	0	0.000	0.00
8	f2_apr_2	f1_apr_2		21.00	25	0	25.000	525.00
9	fact1_1	f1_mar_1	prod f1 19 mar	127.90	500	50	338.333	43272.81
10	f1_apr_1	f1_mar_1	back f1 19 apl	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	prod f1 25 mar	217.90	400	40	400.000	87159.99
13	f1_apr_2	f1_mar_2	back f1 25 apl	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	116.667	10511.68
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	20.000	220.00
22	fact2_1	f2_apr_1	prod f2 19 apl	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	back f2 19 may	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	prod f2 25 apl	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	back f2 25 may	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	prod f2 19 mar	88.00	450	35	290.000	25520.00
31	f2_apr_1	f2_mar_1	back f2 19 apl	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	prod f2 25 mar	182.00	650	35	650.000	118300.00
34	f2_apr_2	f2_mar_2	back f2 25 apl	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	0.000	0.00
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	148.333	-48601.35
42	f1_apr_1	shop1_1		-300.00	250	0	250.000	-75000.00
43	f1_may_1	shop1_1		-285.00	250	0	1.667	-475.01
44	f2_mar_1	shop1_1		-297.40	250	0	250.000	-74350.00
45	f2_apr_1	shop1_1		-290.00	250	0	250.000	-72500.00
46	f2_may_1	shop1_1		-292.00	250	0	0.000	-0.05
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		-515.02	99999999	0	347.500	-178969.34
50	f2_mar_2	shop1_2		-567.83	500	0	500.000	-283914.98
51	f2_apr_2	shop1_2		-542.19	500	0	52.500	-28465.09
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	-90684.99
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	-0.01
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	-283869.90
60	f1_apr_2	shop2_2		-549.68	99999999	0	245.000	-134671.54
61	f1_may_2	shop2_2		-500.00	99999999	0	2.500	-1250.00
62	f2_mar_2	shop2_2		-542.83	500	0	125.000	-67853.77
63	f2_apr_2	shop2_2		-559.19	500	0	500.000	-279594.99
64	f2_may_2	shop2_2		-519.06	500	0	122.500	-63584.94
65			f1 chips from mar	1.00	150	0	20.000	20.00
66			f1 unused chips	0.00	99999999	0	0.001	0.00
67			f2 chips from mar	1.00	150	0	0.000	0.00
68			f2 unused chips	0.00	99999999	0	280.000	0.00
								-1295542.72