Example 4.4: Production, Inventory, Distribution Problem

The NETFLOW Procedure

Example 4.4: Production, Inventory, Distribution Problem

Example 4.4 through Example 4.8 use data from a company that produces two sizes of televisions in order to illustrate variations in the use the NETFLOW procedure. The company makes televisions with a diagonal screen measurement of either 19 inches or 25 inches. These televisions are made between March and May at both of the company's two factories. Each factory has a limit on the total number of televisions of each screen dimension that can be made during those months.

The televisions are distributed to one of two shops, stored at the factory where they were made and sold later, or shipped to the other factory. Some sets can be used to fill back orders from the previous months. Each shop demands a number of each type of TV for the months March through May. The following network in Figure 4.26 illustrates the model. Arc costs can be interpreted as production costs, storage costs, back order penalty costs, inter-factory transportation costs, and sales profits. The arcs can have capacities and lower flow bounds.

Figure 4.26: TV problem

There are two similarly structured networks, one for the 19-inch televisions and the other for the 25-inch screen TVs. The minimum cost production, inventory, and distribution plan for both TV types can be determined in the same run of PROC NETFLOW. To ensure that node names are unambiguous, the names of nodes in the 19-inch network have suffix _1, and the node names in the 25-inch network have suffix _2.

The FUTURE1 option is specified because further processing could be required. Information concerning an optimal solution is retained so it can be used to warm start later optimizations. Warm start information is mostly in variables named _NNUMB_, _PRED_, _TRAV_, _SCESS_, _ARCID_, and _FBQ_ and in observations for nodes named _EXCESS_ and _ROOT_, that are in the NODEOUT=NODE2 output data set. (PROC NETFLOW uses similar devices to store warm start information in the DUALOUT= data set when the FUTURE2 option is specified.) Variables _ANUMB_ and _TNUMB_ and observations for arcs directed from or toward a node called _EXCESS_ are present in ARCOUT=arc1. (PROC NETFLOW uses similar devices to store warm start information in the CONOUT= data set when the FUTURE2 option is specified.)

The following code shows how to save the problem data in data sets and solve the model with PROC NETFLOW:

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   .
;

 proc netflow
        nodedata=node0
        arcdata=arc0;
    set future1
        nodeout=node2
        arcout=arc1;
proc print data=arc1; sum _fcost_;
proc print data=node2;
run;

The following notes 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 iterations performed (neglecting any 
      constraints)= 74 .
NOTE: Of these, 1 were degenerate.
NOTE: Optimum (neglecting any constraints) found.
NOTE: Minimal total cost= -1281110.35 .
NOTE: The data set WORK.ARC1 has 68 observations and 
      18 variables.
NOTE: The data set WORK.NODE2 has 22 observations and 
      10 variables.

The solution is given in the NODEOUT=node2 and ARCOUT=arc1 data sets. In the ARCOUT= data set, shown in Output 4.4.1, the variables diagonal, factory, key_id, and mth_made form an implicit ID list. The diagonal variable has one of two values, 19 or 25. factory also has one of two values, 1 or 2, to denote the factory where either production or storage occurs, from where TVs are either sold to shops or satisfy back orders. PRODUCTION, STORAGE, SALES, and BACKORDER are values of the key_id variable.

Other values of this variable, F1_TO_2 and F2_TO_1, are used when flow through arcs represents the transportation of TVs between factories. The mth_made variable has values MARCH, APRIL, and MAY, the months when TVs that are modeled as flow through an arc were made (assuming that no televisions are stored for more than one month and none manufactured in May are used to fill March back orders).

These ID variables can be used after the PROC NETFLOW run to produce reports and perform analysis on particular parts of the company's operation. For example, reports can be generated for production numbers for each factory; optimal sales figures for each shop; and how many TVs should be stored, used to fill back orders, sent to the other factory, or any combination of these, for TVs with a particular screen, those produced in a particular month, or both.

Output 4.4.1: ARCOUT=ARC1

Minimum Cost Flow problem

Production Planning/Inventory/Distribution

Obs	_tail_	_head_	_cost_	_capac_	_lo_	_name_	_SUPPLY_	_DEMAND_	_FLOW_	_FCOST_	_RCOST_	_ANUMB_	_TNUMB_	_STATUS_	diagonal	factory	key_id	mth_made
1	fact1_1	_EXCESS_	0.00	99999999	0		1000	200	5	0.00	.	65	1	KEY_ARC BASIC	.	.
2	fact2_1	_EXCESS_	0.00	99999999	0		850	200	45	0.00	.	66	10	KEY_ARC BASIC	.	.
3	fact1_2	_EXCESS_	0.00	99999999	0		1000	200	10	0.00	.	67	11	KEY_ARC BASIC	.	.
4	fact2_2	_EXCESS_	0.00	99999999	0		1500	200	140	0.00	.	68	20	KEY_ARC BASIC	.	.
5	fact1_1	f1_apr_1	78.60	600	50	prod f1 19 apl	1000	.	600	47160.00	-0.650	4	1	UPPERBD NONBASIC	19	1	production	April
6	f1_mar_1	f1_apr_1	15.00	50	0		.	.	0	0.00	63.650	5	2	LOWERBD NONBASIC	19	1	storage	March
7	f1_may_1	f1_apr_1	28.00	20	0	back f1 19 may	.	.	0	0.00	43.000	6	4	LOWERBD NONBASIC	19	1	backorder	May
8	f2_apr_1	f1_apr_1	11.00	40	0		.	.	0	0.00	22.000	7	6	LOWERBD NONBASIC	19	.	f2_to_1	April
9	fact1_2	f1_apr_2	174.50	550	50	prod f1 25 apl	1000	.	550	95975.00	-14.350	36	11	UPPERBD NONBASIC	25	1	production	April
10	f1_mar_2	f1_apr_2	20.00	40	0		.	.	0	0.00	94.210	37	12	LOWERBD NONBASIC	25	1	storage	March
11	f1_may_2	f1_apr_2	41.00	15	0	back f1 25 may	.	.	15	615.00	-16.660	38	14	UPPERBD NONBASIC	25	1	backorder	May
12	f2_apr_2	f1_apr_2	21.00	25	0		.	.	0	0.00	30.510	39	16	LOWERBD NONBASIC	25	.	f2_to_1	April
13	fact1_1	f1_mar_1	127.90	500	50	prod f1 19 mar	1000	.	345	44125.50	.	1	1	KEY_ARC BASIC	19	1	production	March
14	f1_apr_1	f1_mar_1	28.00	20	0	back f1 19 apl	.	.	20	560.00	-20.650	2	3	UPPERBD NONBASIC	19	1	backorder	April
15	f2_mar_1	f1_mar_1	10.00	40	0		.	.	40	400.00	-29.900	3	5	UPPERBD NONBASIC	19	.	f2_to_1	March
16	fact1_2	f1_mar_2	217.90	400	40	prod f1 25 mar	1000	.	400	87160.00	-45.160	33	11	UPPERBD NONBASIC	25	1	production	March
17	f1_apr_2	f1_mar_2	32.00	30	0	back f1 25 apl	.	.	30	960.00	-42.210	34	13	UPPERBD NONBASIC	25	1	backorder	April
18	f2_mar_2	f1_mar_2	20.00	25	0		.	.	25	500.00	-61.060	35	15	UPPERBD NONBASIC	25	.	f2_to_1	March
19	fact1_1	f1_may_1	95.10	400	50		1000	.	50	4755.00	0.850	8	1	LOWERBD NONBASIC	19	1	production	May
20	f1_apr_1	f1_may_1	12.00	50	0		.	.	50	600.00	-3.000	9	3	UPPERBD NONBASIC	19	1	storage	April
21	f2_may_1	f1_may_1	13.00	40	0		.	.	0	0.00	29.000	10	7	LOWERBD NONBASIC	19	.	f2_to_1	May
22	fact1_2	f1_may_2	133.30	350	40		1000	.	40	5332.00	2.110	40	11	LOWERBD NONBASIC	25	1	production	May
23	f1_apr_2	f1_may_2	18.00	40	0		.	.	0	0.00	75.660	41	13	LOWERBD NONBASIC	25	1	storage	April
24	f2_may_2	f1_may_2	43.00	25	0		.	.	0	0.00	40.040	42	17	LOWERBD NONBASIC	25	.	f2_to_1	May
25	f1_apr_1	f2_apr_1	11.00	99999999	0		.	.	30	330.00	.	14	3	KEY_ARC BASIC	19	.	f1_to_2	April
26	fact2_1	f2_apr_1	62.40	480	35	prod f2 19 apl	850	.	480	29952.00	-27.850	15	10	UPPERBD NONBASIC	19	2	production	April
27	f2_mar_1	f2_apr_1	18.00	30	0		.	.	0	0.00	15.750	16	5	LOWERBD NONBASIC	19	2	storage	March
28	f2_may_1	f2_apr_1	25.00	15	0	back f2 19 may	.	.	0	0.00	45.000	17	7	LOWERBD NONBASIC	19	2	backorder	May
29	f1_apr_2	f2_apr_2	23.00	99999999	0		.	.	0	0.00	13.490	46	13	LOWERBD NONBASIC	25	.	f1_to_2	April
30	fact2_2	f2_apr_2	196.70	680	35	prod f2 25 apl	1500	.	680	133756.00	-1.660	47	20	UPPERBD NONBASIC	25	2	production	April
31	f2_mar_2	f2_apr_2	28.00	50	0		.	.	0	0.00	11.640	48	15	LOWERBD NONBASIC	25	2	storage	March
32	f2_may_2	f2_apr_2	54.00	15	0	back f2 25 may	.	.	15	810.00	-16.130	49	17	UPPERBD NONBASIC	25	2	backorder	May
33	f1_mar_1	f2_mar_1	11.00	99999999	0		.	.	0	0.00	50.900	11	2	LOWERBD NONBASIC	19	.	f1_to_2	March
34	fact2_1	f2_mar_1	88.00	450	35	prod f2 19 mar	850	.	290	25520.00	.	12	10	KEY_ARC BASIC	19	2	production	March
35	f2_apr_1	f2_mar_1	17.00	15	0	back f2 19 apl	.	.	0	0.00	19.250	13	6	LOWERBD NONBASIC	19	2	backorder	April
36	f1_mar_2	f2_mar_2	23.00	99999999	0		.	.	0	0.00	104.060	43	12	LOWERBD NONBASIC	25	.	f1_to_2	March
37	fact2_2	f2_mar_2	182.00	650	35	prod f2 25 mar	1500	.	645	117390.00	.	44	20	KEY_ARC BASIC	25	2	production	March
38	f2_apr_2	f2_mar_2	31.00	15	0	back f2 25 apl	.	.	0	0.00	47.360	45	16	LOWERBD NONBASIC	25	2	backorder	April
39	f1_may_1	f2_may_1	16.00	99999999	0		.	.	100	1600.00	.	18	4	KEY_ARC BASIC	19	.	f1_to_2	May
40	fact2_1	f2_may_1	133.80	250	35		850	.	35	4683.00	23.550	19	10	LOWERBD NONBASIC	19	2	production	May
41	f2_apr_1	f2_may_1	20.00	30	0		.	.	15	300.00	.	20	6	KEY_ARC BASIC	19	2	storage	April
42	f1_may_2	f2_may_2	26.00	99999999	0		.	.	0	0.00	28.960	50	14	LOWERBD NONBASIC	25	.	f1_to_2	May
43	fact2_2	f2_may_2	201.40	550	35		1500	.	35	7049.00	73.170	51	20	LOWERBD NONBASIC	25	2	production	May
44	f2_apr_2	f2_may_2	38.00	50	0		.	.	0	0.00	108.130	52	16	LOWERBD NONBASIC	25	2	storage	April
45	f1_mar_1	shop1_1	-327.65	250	0		.	900	155	-50785.75	.	21	2	KEY_ARC BASIC	19	1	sales	March
46	f1_apr_1	shop1_1	-300.00	250	0		.	900	250	-75000.00	-21.000	22	3	UPPERBD NONBASIC	19	1	sales	April
47	f1_may_1	shop1_1	-285.00	250	0		.	900	0	0.00	9.000	23	4	LOWERBD NONBASIC	19	1	sales	May
48	f2_mar_1	shop1_1	-297.40	250	0		.	900	250	-74350.00	-9.650	24	5	UPPERBD NONBASIC	19	2	sales	March
49	f2_apr_1	shop1_1	-290.00	250	0		.	900	245	-71050.00	.	25	6	KEY_ARC BASIC	19	2	sales	April
50	f2_may_1	shop1_1	-292.00	250	0		.	900	0	0.00	18.000	26	7	LOWERBD NONBASIC	19	2	sales	May
51	f1_mar_2	shop1_2	-559.76	99999999	0		.	900	0	0.00	47.130	53	12	LOWERBD NONBASIC	25	1	sales	March
52	f1_apr_2	shop1_2	-524.28	99999999	0		.	900	0	0.00	8.400	54	13	LOWERBD NONBASIC	25	1	sales	April
53	f1_may_2	shop1_2	-475.02	99999999	0		.	900	25	-11875.50	.	55	14	KEY_ARC BASIC	25	1	sales	May
54	f2_mar_2	shop1_2	-567.83	500	0		.	900	500	-283915.00	-42.000	56	15	UPPERBD NONBASIC	25	2	sales	March
55	f2_apr_2	shop1_2	-542.19	500	0		.	900	375	-203321.25	.	57	16	KEY_ARC BASIC	25	2	sales	April
56	f2_may_2	shop1_2	-461.56	500	0		.	900	0	0.00	10.500	58	17	LOWERBD NONBASIC	25	2	sales	May
57	f1_mar_1	shop2_1	-362.74	250	0		.	900	250	-90685.00	-46.090	27	2	UPPERBD NONBASIC	19	1	sales	March
58	f1_apr_1	shop2_1	-300.00	250	0		.	900	250	-75000.00	-32.000	28	3	UPPERBD NONBASIC	19	1	sales	April
59	f1_may_1	shop2_1	-245.00	250	0		.	900	0	0.00	38.000	29	4	LOWERBD NONBASIC	19	1	sales	May
60	f2_mar_1	shop2_1	-272.70	250	0		.	900	0	0.00	4.050	30	5	LOWERBD NONBASIC	19	2	sales	March
61	f2_apr_1	shop2_1	-312.00	250	0		.	900	250	-78000.00	-33.000	31	6	UPPERBD NONBASIC	19	2	sales	April
62	f2_may_1	shop2_1	-299.00	250	0		.	900	150	-44850.00	.	32	7	KEY_ARC BASIC	19	2	sales	May
63	f1_mar_2	shop2_2	-623.89	99999999	0		.	1450	455	-283869.95	.	59	12	KEY_ARC BASIC	25	1	sales	March
64	f1_apr_2	shop2_2	-549.68	99999999	0		.	1450	535	-294078.80	.	60	13	KEY_ARC BASIC	25	1	sales	April
65	f1_may_2	shop2_2	-460.00	99999999	0		.	1450	0	0.00	32.020	61	14	LOWERBD NONBASIC	25	1	sales	May
66	f2_mar_2	shop2_2	-542.83	500	0		.	1450	120	-65139.60	.	62	15	KEY_ARC BASIC	25	2	sales	March
67	f2_apr_2	shop2_2	-559.19	500	0		.	1450	320	-178940.80	.	63	16	KEY_ARC BASIC	25	2	sales	April
68	f2_may_2	shop2_2	-489.06	500	0		.	1450	20	-9781.20	.	64	17	KEY_ARC BASIC	25	2	sales	May
										-1281110.35

Output 4.4.2: NODEOUT=NODE2

Obs	_node_	_supdem_	_DUAL_	_NNUMB_	_PRED_	_TRAV_	_SCESS_	_ARCID_	_FLOW_	_FBQ_
1	_ROOT_	238	0.00	22	0	8	0	3	166	-69
2	_EXCESS_	-200	-100000198.75	21	1	11	13	65	5	65
3	f1_apr_1	.	-100000278.00	3	6	7	1	-14	30	4
4	f1_apr_2	.	-100000387.60	13	19	17	1	-60	535	36
5	f1_mar_1	.	-100000326.65	2	8	1	15	-21	155	1
6	f1_mar_2	.	-100000461.81	12	19	13	1	-59	455	33
7	f1_may_1	.	-100000293.00	4	7	2	1	-18	100	8
8	f1_may_2	.	-100000329.94	14	18	12	1	-55	25	40
9	f2_apr_1	.	-100000289.00	6	8	3	5	-25	245	14
10	f2_apr_2	.	-100000397.11	16	19	18	3	-63	320	46
11	f2_mar_1	.	-100000286.75	5	10	22	1	12	255	11
12	f2_mar_2	.	-100000380.75	15	20	19	8	44	610	43
13	f2_may_1	.	-100000309.00	7	6	9	3	20	15	18
14	f2_may_2	.	-100000326.98	17	19	10	1	-64	20	50
15	fact1_1	1000	-100000198.75	1	2	21	14	-1	295	-1
16	fact1_2	1000	-100000198.75	11	21	20	1	-67	10	-33
17	fact2_1	850	-100000198.75	10	21	5	2	-66	45	-33
18	fact2_2	1500	-100000198.75	20	21	15	9	-68	140	-65
19	shop1_1	-900	-99999999.00	8	22	6	21	0	0	21
20	shop1_2	-900	-99999854.92	18	16	14	2	57	375	53
21	shop2_1	-900	-100000010.00	9	7	4	1	32	150	27
22	shop2_2	-1450	-99999837.92	19	15	16	7	62	120	59

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