The Linear Programming Solver

Example 5.8 Migration to OPTMODEL: Production, Inventory, Distribution

The following example shows how to use PROC OPTMODEL to solve the example "Production, Inventory, Distribution Problem" in Chapter 6, The NETFLOW Procedure (SAS/OR User's Guide: Mathematical Programming Legacy Procedures). The input data sets are the same as in that example.

title 'Minimum Cost Flow Problem';
title2 'Production Planning/Inventory/Distribution';

data node0;
   input _node_ $ _supdem_ ;
   datalines;
fact1_1 1000
fact2_1 850
fact1_2 1000
fact2_2 1500
shop1_1 -900
shop2_1 -900
shop1_2 -900
shop2_2 -1450
;

data arc0;
   input _tail_ $ _head_ $ _cost_ _capac_ _lo_ 
      diagonal factory key_id $10. mth_made $ _name_&$17.;
   datalines;
fact1_1  f1_mar_1 127.9   500 50 19 1 production March prod f1 19 mar
fact1_1  f1_apr_1 78.6    600 50 19 1 production April prod f1 19 apl
fact1_1  f1_may_1 95.1    400 50 19 1 production May   .
f1_mar_1 f1_apr_1 15      50  .  19 1 storage    March .
f1_apr_1 f1_may_1 12      50  .  19 1 storage    April .
f1_apr_1 f1_mar_1 28      20  .  19 1 backorder  April back f1 19 apl
f1_may_1 f1_apr_1 28      20  .  19 1 backorder  May   back f1 19 may
f1_mar_1 f2_mar_1 11      .   .  19 . f1_to_2    March .
f1_apr_1 f2_apr_1 11      .   .  19 . f1_to_2    April .
f1_may_1 f2_may_1 16      .   .  19 . f1_to_2    May   .
f1_mar_1 shop1_1  -327.65 250 .  19 1 sales      March .
f1_apr_1 shop1_1  -300    250 .  19 1 sales      April .
f1_may_1 shop1_1  -285    250 .  19 1 sales      May   .
f1_mar_1 shop2_1  -362.74 250 .  19 1 sales      March .
f1_apr_1 shop2_1  -300    250 .  19 1 sales      April .
f1_may_1 shop2_1  -245    250 .  19 1 sales      May   .
fact2_1  f2_mar_1 88.0    450 35 19 2 production March prod f2 19 mar
fact2_1  f2_apr_1 62.4    480 35 19 2 production April prod f2 19 apl
fact2_1  f2_may_1 133.8   250 35 19 2 production May   .
f2_mar_1 f2_apr_1 18      30  .  19 2 storage    March .
f2_apr_1 f2_may_1 20      30  .  19 2 storage    April .
f2_apr_1 f2_mar_1 17      15  .  19 2 backorder  April back f2 19 apl
f2_may_1 f2_apr_1 25      15  .  19 2 backorder  May   back f2 19 may
f2_mar_1 f1_mar_1 10      40  .  19 . f2_to_1    March .
f2_apr_1 f1_apr_1 11      40  .  19 . f2_to_1    April .
f2_may_1 f1_may_1 13      40  .  19 . f2_to_1    May   .
f2_mar_1 shop1_1  -297.4  250 .  19 2 sales      March .
f2_apr_1 shop1_1  -290    250 .  19 2 sales      April .
f2_may_1 shop1_1  -292    250 .  19 2 sales      May   .
f2_mar_1 shop2_1  -272.7  250 .  19 2 sales      March .
f2_apr_1 shop2_1  -312    250 .  19 2 sales      April .
f2_may_1 shop2_1  -299    250 .  19 2 sales      May   .
fact1_2  f1_mar_2 217.9   400 40 25 1 production March prod f1 25 mar
fact1_2  f1_apr_2 174.5   550 50 25 1 production April prod f1 25 apl
fact1_2  f1_may_2 133.3   350 40 25 1 production May   .
f1_mar_2 f1_apr_2 20      40  .  25 1 storage    March .
f1_apr_2 f1_may_2 18      40  .  25 1 storage    April .
f1_apr_2 f1_mar_2 32      30  .  25 1 backorder  April back f1 25 apl
f1_may_2 f1_apr_2 41      15  .  25 1 backorder  May   back f1 25 may
f1_mar_2 f2_mar_2 23      .   .  25 . f1_to_2    March .
f1_apr_2 f2_apr_2 23      .   .  25 . f1_to_2    April .
f1_may_2 f2_may_2 26      .   .  25 . f1_to_2    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  -475.02 .   .  25 1 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  -460.00 .   .  25 1 sales      May   .
fact2_2  f2_mar_2 182.0   650 35 25 2 production March prod f2 25 mar
fact2_2  f2_apr_2 196.7   680 35 25 2 production April prod f2 25 apl
fact2_2  f2_may_2 201.4   550 35 25 2 production May   .
f2_mar_2 f2_apr_2 28      50  .  25 2 storage    March .
f2_apr_2 f2_may_2 38      50  .  25 2 storage    April .
f2_apr_2 f2_mar_2 31      15  .  25 2 backorder  April back f2 25 apl
f2_may_2 f2_apr_2 54      15  .  25 2 backorder  May   back f2 25 may
f2_mar_2 f1_mar_2 20      25  .  25 . f2_to_1    March .
f2_apr_2 f1_apr_2 21      25  .  25 . f2_to_1    April .
f2_may_2 f1_may_2 43      25  .  25 . f2_to_1    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  -461.56 500 .  25 2 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  -489.06 500 .  25 2 sales      May   .
;

The following PROC OPTMODEL statements read the data sets, build the linear programming model, solve the model, and output the optimal solution to SAS data sets called ARC1 and NODE2:

proc optmodel;
   set <str> NODES;
   num _supdem_ {NODES} init 0;
   read data node0 into NODES=[_node_] _supdem_;

   set <str,str> ARCS;
   num _lo_ {ARCS} init 0;
   num _capac_ {ARCS} init .;
   num _cost_ {ARCS};
   num diagonal {ARCS};
   num factory {ARCS};
   str key_id {ARCS};
   str mth_made {ARCS};
   str _name_ {ARCS};
   read data arc0 nomiss into ARCS=[_tail_ _head_] _lo_ _capac_ _cost_ 
      diagonal factory key_id mth_made _name_;
   NODES = NODES union (union {<i,j> in ARCS} {i,j});

   var Flow {<i,j> in ARCS} >= _lo_[i,j];
   for {<i,j> in ARCS: _capac_[i,j] ne .} Flow[i,j].ub = _capac_[i,j];
   min obj = sum {<i,j> in ARCS} _cost_[i,j] * Flow[i,j];
   con balance {i in NODES}: sum {<(i),j> in ARCS} Flow[i,j] 
      - sum {<j,(i)> in ARCS} Flow[j,i] = _supdem_[i];

   num infinity = min {r in {}} r;
   num excess = sum {i in NODES} _supdem_[i];
   if (excess > 0) then do;
      /* change equality constraint to le constraint for supply nodes */
      for {i in NODES: _supdem_[i] > 0} balance[i].lb = -infinity;
   end;
   else if (excess < 0) then do;
      /* change equality constraint to ge constraint for demand nodes */
      for {i in NODES: _supdem_[i] < 0} balance[i].ub = infinity;
   end;

   solve;

   num _supply_ {<i,j> in ARCS} = 
      (if _supdem_[i] ne 0 then _supdem_[i] else .); 
   num _demand_ {<i,j> in ARCS} = 
      (if _supdem_[j] ne 0 then -_supdem_[j] else .); 
   num _fcost_ {<i,j> in ARCS} = _cost_[i,j] * Flow[i,j].sol;

   create data arc1 from [_tail_ _head_]
      _cost_ _capac_ _lo_ _name_ _supply_ _demand_ _flow_=Flow _fcost_
      _rcost_ = 
         (if Flow[_tail_,_head_].rc ne 0 then Flow[_tail_,_head_].rc else .)
      _status_ = Flow.status diagonal factory key_id mth_made;
   create data node2 from [_node_]
      _supdem_ = (if _supdem_[_node_] ne 0 then _supdem_[_node_] else .)
      _dual_ = balance.dual;
quit;

The PROC OPTMODEL statements use both single-dimensional (NODES) and multiple-dimensional (ARCS) index sets, which are populated from the corresponding data set variables in the READ DATA statements. The _SUPDEM_, _LO_, and _CAPAC_ parameters are given initial values, and the NOMISS option in the READ DATA statement tells PROC OPTMODEL to read only the nonmissing values from the input data set. The balance constraint is initially declared as an equality, but depending on the total supply or demand, the sense of this constraint is changed to " $\text{[math]}$ " or " $\text{[math]}$ " by relaxing the constraint’s lower or upper bound, respectively. The ARC1 output data set contains most of the same information as in the NETFLOW example, including reduced cost, basis status, and dual values. The _ANUMB_ and _TNUMB_ values do not apply here.

The PROC PRINT statements are similar to the PROC NETFLOW example:

options ls=80 ps=54;
proc print data=arc1 heading=h width=min;
   var _tail_ _head_ _cost_ _capac_ _lo_ _name_ 
   _supply_ _demand_ _flow_ _fcost_;
   sum _fcost_;
run;
proc print data=arc1 heading=h width=min;
   var _rcost_ _status_ diagonal factory key_id mth_made;
run;
proc print data=node2;
run;

The output data sets are displayed in Output 5.8.1.

Output 5.8.1 Output Data Sets

Minimum Cost Flow Problem

Production Planning/Inventory/Distribution

Obs	_tail_	_head_	_cost_	_capac_	_lo_	_name_	_supply_	_demand_	_flow_	_fcost_
1	fact1_1	f1_mar_1	127.90	500	50	prod f1 19 mar	1000	.	345	44125.50
2	fact1_1	f1_apr_1	78.60	600	50	prod f1 19 apl	1000	.	600	47160.00
3	fact1_1	f1_may_1	95.10	400	50		1000	.	50	4755.00
4	f1_mar_1	f1_apr_1	15.00	50	0		.	.	0	0.00
5	f1_apr_1	f1_may_1	12.00	50	0		.	.	50	600.00
6	f1_apr_1	f1_mar_1	28.00	20	0	back f1 19 apl	.	.	20	560.00
7	f1_may_1	f1_apr_1	28.00	20	0	back f1 19 may	.	.	0	0.00
8	f1_mar_1	f2_mar_1	11.00	.	0		.	.	0	0.00
9	f1_apr_1	f2_apr_1	11.00	.	0		.	.	30	330.00
10	f1_may_1	f2_may_1	16.00	.	0		.	.	100	1600.00
11	f1_mar_1	shop1_1	-327.65	250	0		.	900	155	-50785.75
12	f1_apr_1	shop1_1	-300.00	250	0		.	900	250	-75000.00
13	f1_may_1	shop1_1	-285.00	250	0		.	900	0	0.00
14	f1_mar_1	shop2_1	-362.74	250	0		.	900	250	-90685.00
15	f1_apr_1	shop2_1	-300.00	250	0		.	900	250	-75000.00
16	f1_may_1	shop2_1	-245.00	250	0		.	900	0	0.00
17	fact2_1	f2_mar_1	88.00	450	35	prod f2 19 mar	850	.	290	25520.00
18	fact2_1	f2_apr_1	62.40	480	35	prod f2 19 apl	850	.	480	29952.00
19	fact2_1	f2_may_1	133.80	250	35		850	.	35	4683.00
20	f2_mar_1	f2_apr_1	18.00	30	0		.	.	0	0.00
21	f2_apr_1	f2_may_1	20.00	30	0		.	.	15	300.00
22	f2_apr_1	f2_mar_1	17.00	15	0	back f2 19 apl	.	.	0	0.00
23	f2_may_1	f2_apr_1	25.00	15	0	back f2 19 may	.	.	0	0.00
24	f2_mar_1	f1_mar_1	10.00	40	0		.	.	40	400.00
25	f2_apr_1	f1_apr_1	11.00	40	0		.	.	0	0.00
26	f2_may_1	f1_may_1	13.00	40	0		.	.	0	0.00
27	f2_mar_1	shop1_1	-297.40	250	0		.	900	250	-74350.00
28	f2_apr_1	shop1_1	-290.00	250	0		.	900	245	-71050.00
29	f2_may_1	shop1_1	-292.00	250	0		.	900	0	0.00
30	f2_mar_1	shop2_1	-272.70	250	0		.	900	0	0.00
31	f2_apr_1	shop2_1	-312.00	250	0		.	900	250	-78000.00
32	f2_may_1	shop2_1	-299.00	250	0		.	900	150	-44850.00
33	fact1_2	f1_mar_2	217.90	400	40	prod f1 25 mar	1000	.	400	87160.00
34	fact1_2	f1_apr_2	174.50	550	50	prod f1 25 apl	1000	.	550	95975.00
35	fact1_2	f1_may_2	133.30	350	40		1000	.	40	5332.00
36	f1_mar_2	f1_apr_2	20.00	40	0		.	.	0	0.00
37	f1_apr_2	f1_may_2	18.00	40	0		.	.	0	0.00
38	f1_apr_2	f1_mar_2	32.00	30	0	back f1 25 apl	.	.	30	960.00
39	f1_may_2	f1_apr_2	41.00	15	0	back f1 25 may	.	.	15	615.00
40	f1_mar_2	f2_mar_2	23.00	.	0		.	.	0	0.00
41	f1_apr_2	f2_apr_2	23.00	.	0		.	.	0	0.00
42	f1_may_2	f2_may_2	26.00	.	0		.	.	0	0.00
43	f1_mar_2	shop1_2	-559.76	.	0		.	900	0	0.00
44	f1_apr_2	shop1_2	-524.28	.	0		.	900	0	0.00
45	f1_may_2	shop1_2	-475.02	.	0		.	900	25	-11875.50
46	f1_mar_2	shop2_2	-623.89	.	0		.	1450	455	-283869.95
47	f1_apr_2	shop2_2	-549.68	.	0		.	1450	535	-294078.80
48	f1_may_2	shop2_2	-460.00	.	0		.	1450	0	0.00
49	fact2_2	f2_mar_2	182.00	650	35	prod f2 25 mar	1500	.	645	117390.00
50	fact2_2	f2_apr_2	196.70	680	35	prod f2 25 apl	1500	.	680	133756.00
51	fact2_2	f2_may_2	201.40	550	35		1500	.	35	7049.00
52	f2_mar_2	f2_apr_2	28.00	50	0		.	.	0	0.00
53	f2_apr_2	f2_may_2	38.00	50	0		.	.	0	0.00
54	f2_apr_2	f2_mar_2	31.00	15	0	back f2 25 apl	.	.	0	0.00
55	f2_may_2	f2_apr_2	54.00	15	0	back f2 25 may	.	.	15	810.00
56	f2_mar_2	f1_mar_2	20.00	25	0		.	.	25	500.00
57	f2_apr_2	f1_apr_2	21.00	25	0		.	.	0	0.00
58	f2_may_2	f1_may_2	43.00	25	0		.	.	0	0.00
59	f2_mar_2	shop1_2	-567.83	500	0		.	900	500	-283915.00
60	f2_apr_2	shop1_2	-542.19	500	0		.	900	375	-203321.25
61	f2_may_2	shop1_2	-461.56	500	0		.	900	0	0.00
62	f2_mar_2	shop2_2	-542.83	500	0		.	1450	120	-65139.60
63	f2_apr_2	shop2_2	-559.19	500	0		.	1450	320	-178940.80
64	f2_may_2	shop2_2	-489.06	500	0		.	1450	20	-9781.20
										-1281110.35

Minimum Cost Flow Problem

Production Planning/Inventory/Distribution

Obs	_rcost_	_status_	diagonal	factory	key_id	mth_made
1	.	B	19	1	production	March
2	-0.65	U	19	1	production	April
3	0.85	L	19	1	production	May
4	63.65	L	19	1	storage	March
5	-3.00	U	19	1	storage	April
6	-20.65	U	19	1	backorder	April
7	43.00	L	19	1	backorder	May
8	50.90	L	19	.	f1_to_2	March
9	.	B	19	.	f1_to_2	April
10	.	B	19	.	f1_to_2	May
11	.	B	19	1	sales	March
12	-21.00	U	19	1	sales	April
13	9.00	L	19	1	sales	May
14	-46.09	U	19	1	sales	March
15	-32.00	U	19	1	sales	April
16	38.00	L	19	1	sales	May
17	.	B	19	2	production	March
18	-27.85	U	19	2	production	April
19	23.55	L	19	2	production	May
20	15.75	L	19	2	storage	March
21	.	B	19	2	storage	April
22	19.25	L	19	2	backorder	April
23	45.00	L	19	2	backorder	May
24	-29.90	U	19	.	f2_to_1	March
25	22.00	L	19	.	f2_to_1	April
26	29.00	L	19	.	f2_to_1	May
27	-9.65	U	19	2	sales	March
28	.	B	19	2	sales	April
29	18.00	L	19	2	sales	May
30	4.05	L	19	2	sales	March
31	-33.00	U	19	2	sales	April
32	.	B	19	2	sales	May
33	-45.16	U	25	1	production	March
34	-14.35	U	25	1	production	April
35	2.11	L	25	1	production	May
36	94.21	L	25	1	storage	March
37	75.66	L	25	1	storage	April
38	-42.21	U	25	1	backorder	April
39	-16.66	U	25	1	backorder	May
40	104.06	L	25	.	f1_to_2	March
41	13.49	L	25	.	f1_to_2	April
42	28.96	L	25	.	f1_to_2	May
43	47.13	L	25	1	sales	March
44	8.40	L	25	1	sales	April
45	.	B	25	1	sales	May
46	.	B	25	1	sales	March
47	.	B	25	1	sales	April
48	32.02	L	25	1	sales	May
49	.	B	25	2	production	March
50	-1.66	U	25	2	production	April
51	73.17	L	25	2	production	May
52	11.64	L	25	2	storage	March
53	108.13	L	25	2	storage	April
54	47.36	L	25	2	backorder	April
55	-16.13	U	25	2	backorder	May
56	-61.06	U	25	.	f2_to_1	March
57	30.51	L	25	.	f2_to_1	April
58	40.04	L	25	.	f2_to_1	May
59	-42.00	U	25	2	sales	March
60	.	B	25	2	sales	April
61	10.50	L	25	2	sales	May
62	.	B	25	2	sales	March
63	.	B	25	2	sales	April
64	.	B	25	2	sales	May

Minimum Cost Flow Problem

Production Planning/Inventory/Distribution

Obs	_node_	_supdem_	_dual_
1	fact1_1	1000	0.00
2	fact2_1	850	0.00
3	fact1_2	1000	0.00
4	fact2_2	1500	0.00
5	shop1_1	-900	199.75
6	shop2_1	-900	188.75
7	shop1_2	-900	343.83
8	shop2_2	-1450	360.83
9	f1_mar_1	.	-127.90
10	f1_apr_1	.	-79.25
11	f1_may_1	.	-94.25
12	f2_mar_1	.	-88.00
13	f2_apr_1	.	-90.25
14	f2_may_1	.	-110.25
15	f1_mar_2	.	-263.06
16	f1_apr_2	.	-188.85
17	f1_may_2	.	-131.19
18	f2_mar_2	.	-182.00
19	f2_apr_2	.	-198.36
20	f2_may_2	.	-128.23

The log is displayed in Output 5.8.2.

Output 5.8.2 OPTMODEL Log

Minimum Cost Flow Problem

Production Planning/Inventory/Distribution

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

NOTE: There were 64 observations read from the data set WORK.ARC0.

NOTE: The problem has 64 variables (0 free, 0 fixed).

NOTE: The problem has 20 linear constraints (4 LE, 16 EQ, 0 GE, 0 range).

NOTE: The problem has 128 linear constraint coefficients.

NOTE: The problem has 0 nonlinear constraints (0 LE, 0 EQ, 0 GE, 0 range).

NOTE: The OPTMODEL presolver is disabled for linear problems.

NOTE: The OPTLP presolver value AUTOMATIC is applied.

NOTE: The OPTLP presolver removed 0 variables and 0 constraints.

NOTE: The OPTLP presolver removed 0 constraint coefficients.

NOTE: The presolved problem has 64 variables, 20 constraints, and 128

constraint coefficients.

NOTE: The DUAL SIMPLEX solver is called.

Objective

Phase Iteration Value

2 1 -2952213

2 30 -1281110

NOTE: Optimal.

NOTE: Objective = -1281110.35.

NOTE: The data set WORK.ARC1 has 64 observations and 16 variables.

NOTE: The data set WORK.NODE2 has 20 observations and 3 variables.