Example 6.2: Capacitated Transportation Network

The TRANS Procedure

Example 6.2: Capacitated Transportation Network

In this example, the optimal flow is found on a capacitated transportation network. Suppose that there are upper bounds on the amount that can be shipped within each city. The following SAS program and output show how this capacity constraint is included in the model:

   title 'Capacitated Transportation Network';

   data capcty;
      input Atlanta Chicago Denver Houston Los_Ange Miami
         New_York San_Fran Seattle Washingt city$;
      datalines;
   10  .   .  .  .  .  .  .  .  . Atlanta
    . 60   .  .  .  .  .  .  .  . Chicago
    .  . 100  .  .  .  .  .  .  . Denver
    .  .   . 10  .  .  .  .  .  . Houston
    .  .   .  . 30  .  .  .  .  . Los_Ange
    .  .   .  .  . 20  .  .  .  . Miami
    .  .   .  .  .  . 75  .  .  . New_York
    .  .   .  .  .  .  . 25  .  . San_Fran
    .  .   .  .  .  .  .  . 10  . Seattle
    .  .   .  .  .  .  .  .  . 10 Washingt
   ;

   proc trans cost=cst capacity=capcty;
      HEADNODE Atlanta--Washingt;
      TAILNODE city;
      supply supply;
   run;

   proc print;
   run;

After this program executes, the following message is written to the SAS log:

   NOTE: Optimal Solution Total = 24036.

The preceding statements produce the SAS data set in Output 6.2.1

Output 6.2.1: Capacitated Transportation Network Solution

Capacitated Transportation Network with MINFLOW

Obs	city	supply	Atlanta	Chicago	Denver	Houston	Los_Ange	Miami	New_York	San_Fran	Seattle	Washingt	_DUAL_
1	_DEMAND_	.	50	75	89	8	27	39	64	100	50	8	.
2	Atlanta	10	0	0	0	0	0	10	0	0	0	0	-53
3	Chicago	150	44	60	3	0	0	0	0	0	43	0	-2
4	Denver	90	0	0	86	0	0	0	0	4	0	0	-74
5	Houston	27	0	0	0	8	18	1	0	0	0	0	-17
6	Los_Ange	80	0	0	0	0	9	0	0	71	0	0	-134
7	Miami	26	6	0	0	0	0	20	0	0	0	0	0
8	New_York	80	0	8	0	0	0	0	64	0	0	8	-9
9	San_Fran	25	0	0	0	0	0	0	0	25	0	0	-148
10	Seattle	7	0	0	0	0	0	0	0	0	7	0	-155
11	Washingt	15	0	7	0	0	0	8	0	0	0	0	-21
12	_DUAL_	.	60	80	94	37	154	113	29	168	175	29	.

Note that the optimal objective value is greater in the capacitated network (24036) than in the uncapacitated network (22928). Additional constraints can never decrease the objective value of a minimization problem at optimality. Also observe that the flow within Chicago, Miami, and San_Fran are at their limits. The rerouting of flow within these cities accounts for the increase in cost.

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