Consider the following transshipment problem for an oil company. Crude oil is shipped to refineries where it is processed into gasoline and diesel fuel. The gasoline and diesel fuel are then distributed to service stations. At each stage, there are shipping, processing, and distribution costs. Also, there are lower flow bounds and capacities.

In addition, there are two sets of side constraints. The first set is that two times the crude from the Middle East cannot exceed the throughput of a refinery plus 15 units. (The phrase “plus 15 units” that finishes the last sentence is used to enable some side constraints in this example to have a nonzero rhs.) The second set of constraints are necessary to model the situation that one unit of crude mix processed at a refinery yields three-fourths of a unit of gasoline and one-fourth of a unit of diesel fuel.

Because there are two products that are not independent in the way in which they flow through the network, an NPSC is an appropriate model for this example (see Figure 4.6). The side constraints are used to model the limitations on the amount of Middle Eastern crude that can be processed by each refinery and the conversion proportions of crude to gasoline and diesel fuel.

To solve this problem with PROC INTPOINT, save a representation of the model in three SAS data sets. In the NODEDATA= data set, you name the supply and demand nodes and give the associated supplies and demands. To distinguish demand nodes from supply nodes, specify demands as negative quantities. For the oil example, the NODEDATA= data set can be saved as follows:

title  'Oil Industry Example';
title3 'Setting Up Nodedata = Noded For PROC INTPOINT';
data noded;
   input   _node_&$15. _sd_;
   datalines;
middle east       100
u.s.a.             80
servstn1 gas      -95
servstn1 diesel   -30
servstn2 gas      -40
servstn2 diesel   -15
;

The ARCDATA= data set contains the rest of the information about the network. Each observation in the data set identifies an arc in the network and gives the cost per flow unit across the arc, the capacities of the arc, the lower bound on flow across the arc, and the name of the arc.

title3 'Setting Up Arcdata = Arcd1 For PROC INTPOINT';
data arcd1;
   input _from_&$11. _to_&$15. _cost_ _capac_ _lo_ _name_ $;
   datalines;
middle east    refinery 1        63     95   20    m_e_ref1
middle east    refinery 2        81     80   10    m_e_ref2
u.s.a.         refinery 1        55      .    .    .
u.s.a.         refinery 2        49      .    .    .
refinery 1     r1               200    175   50    thruput1
refinery 2     r2               220    100   35    thruput2
r1             ref1 gas           .    140    .    r1_gas
r1             ref1 diesel        .     75    .    .
r2             ref2 gas           .    100    .    r2_gas
r2             ref2 diesel        .     75    .    .
ref1 gas       servstn1 gas      15     70    .    .
ref1 gas       servstn2 gas      22     60    .    .
ref1 diesel    servstn1 diesel   18      .    .    .
ref1 diesel    servstn2 diesel   17      .    .    .
ref2 gas       servstn1 gas      17     35    5    .
ref2 gas       servstn2 gas      31      .    .    .
ref2 diesel    servstn1 diesel   36      .    .    .
ref2 diesel    servstn2 diesel   23      .    .    .
;

Finally, the CONDATA= data set contains the side constraints for the model:

title3 'Setting Up Condata = Cond1 For PROC INTPOINT';
data cond1;
   input m_e_ref1 m_e_ref2 thruput1 r1_gas thruput2 r2_gas
         _type_ $ _rhs_;
   datalines;
-2  .  1 .  . . >= -15
 . -2  . .  1 . GE -15
 .  . -3 4  . . EQ   0
 .  .  . . -3 4  =   0
;

Note that the SAS variable names in the CONDATA= data set are the names of arcs given in the ARCDATA= data set. These are the arcs that have nonzero constraint coefficients in side constraints. For example, the proportionality constraint that specifies that one unit of crude at each refinery yields three-fourths of a unit of gasoline and one-fourth of a unit of diesel fuel is given for refinery 1 in the third observation and for refinery 2 in the last observation. The third observation requires that each unit of flow on the arc thruput1 equals three-fourths of a unit of flow on the arc r1_gas. Because all crude processed at refinery 1 flows through thruput1 and all gasoline produced at refinery 1 flows through r1_gas, the constraint models the situation. It proceeds similarly for refinery 2 in the last observation.

To find the minimum cost flow through the network that satisfies the supplies, demands, and side constraints, invoke PROC INTPOINT as follows:

proc intpoint
   bytes=1000000
   nodedata=noded        /* the supply and demand data */
   arcdata=arcd1         /* the arc descriptions       */
   condata=cond1         /* the side constraints       */
   conout=solution;      /* the solution data set      */
   run;

The following messages, which appear on the SAS log, summarize the model as read by PROC INTPOINT and note the progress toward a solution.

The first set of messages shows the size of the problem. The next set of messages provides statistics on the size of the equivalent LP problem. The number of variables may not equal the number of arcs if the problem has nonarc variables. This example has none. To convert a network to the equivalent LP problem, a flow conservation constraint must be created for each node (including an excess or bypass node, if required). This explains why the number of equality constraints and the number of constraint coefficients differ from the number of equality side constraints and the number of coefficients in all side constraints.

If the preprocessor was successful in decreasing the problem size, some messages will report how well it did. In this example, the model size was cut approximately in half!

The next set of messages describes aspects of the interior point algorithm. Of particular interest are those concerned with the Cholesky factorization of where is the coefficient matrix of the final LP. It is crucial to preorder the rows and columns of this matrix to prevent fill-in and reduce the number of row operations to undertake the factorization. See the section Interior Point Algorithmic Details for a more extensive explanation.

Unlike PROC LP, which displays the solution and other information as output, PROC INTPOINT saves the optimum in the output SAS data set that you specify. For this example, the solution is saved in the SOLUTION data set. It can be displayed with the PRINT procedure as

title3 'Optimum'; 
proc print data=solution;
   var _from_ _to_ _cost_ _capac_ _lo_ _name_ 
       _supply_ _demand_ _flow_ _fcost_;
   sum _fcost_;
   run;

Notice that, in CONOUT=SOLUTION (Figure 4.7), the optimal flow through each arc in the network is given in the variable named _FLOW_, and the cost of flow through each arc is given in the variable _FCOST_.