Missing S Supply and Missing D Demand Values

The INTPOINT Procedure

Missing S Supply and Missing D Demand Values

In some models, you may want a node to be either a supply or demand node but you want the node to supply or demand the optimal number of flow units. To indicate that a node is such a supply node, use a missing S value in the SUPPLY list variable in the ARCDATA= data set or the SUPDEM list variable in the NODEDATA= data set. To indicate that a node is such a demand node, use a missing D value in the DEMAND list variable in the ARCDATA= data set or the SUPDEM list variable in the NODEDATA= data set. Suppose the oil example in the "Introductory NPSC Example" section is changed so that crude oil can be obtained from either the Middle East or U.S.A. in any amounts. You should specify that the node middle east is a supply node, but you do not want to stipulate that it supplies 100 units, as before. The node u.s.a. should also remain a supply node, but you do not want to stipulate that it supplies 80 units. You must specify that these nodes have missing S supply capabilities:

   title  'Oil Industry Example';
   title3 'Crude Oil can come from anywhere';
   data miss_s;
      missing S;
      input   _node_&$15. _sd_;
      datalines;
   middle east         S
   u.s.a.              S
   servstn1 gas      -95
   servstn1 diesel   -30
   servstn2 gas      -40
   servstn2 diesel   -15
   ;

The following PROC INTPOINT run uses the same ARCDATA= and CONDATA= data sets used in the "Introductory NPSC Example" section:

   proc intpoint
      bytes=100000
      nodedata=miss_s       /* the supply (missing S) and */
                            /* demand data                */
      arcdata=arcd1         /* the arc descriptions       */
      condata=cond1         /* the side constraints       */
      conout=solution;      /* the solution data set      */
   run;
   proc print;
      var _from_ _to_ _cost_ _capac_ _lo_ _flow_ _fcost_;
      sum _fcost_;
   run;

The following messages appear on the SAS log:

   NOTE: Number of nodes= 14 .
   NOTE: Number of supply nodes= 2 .
   NOTE: Of these, 2 have unspecified (.S) supply capability.
   NOTE: Number of demand nodes= 4 .
   NOTE: Total supply= 0 , total demand= 180 .
   NOTE: Number of arcs= 18 .
   NOTE: Number of <= side constraints= 0 .
   NOTE: Number of == side constraints= 2 .
   NOTE: Number of >= side constraints= 2 .
   NOTE: Number of side constraint coefficients= 8 .
   NOTE: The following messages relate to the equivalent Linear Program solved by the Interior Point algorithm.
   NOTE: Number of <= constraints= 0 .
   NOTE: Number of == constraints= 17 .
   NOTE: Number of >= constraints= 2 .
   NOTE: Number of constraint coefficients= 48 .
   NOTE: Number of variables= 20 .
   NOTE: After preprocessing, number of <= constraints= 0.
   NOTE: After preprocessing, number of == constraints= 8.
   NOTE: After preprocessing, number of >= constraints= 2.
   NOTE: The preprocessor eliminated 9 constraints from the problem.
   NOTE: The preprocessor eliminated 20 constraint coefficients from the problem.
   NOTE: After preprocessing, number of variables= 11.
   NOTE: The preprocessor eliminated 9 variables 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 21 nonzero elements in A * A transpose.
   NOTE: Of the 10 rows and columns, 4 are sparse.
   NOTE: There are 15 nonzero superdiagonal elements in the sparse rows of the factored A * A transpose. This includes fill-in.
   NOTE: There are 5 operations of the form u[i,j]=u[i,j]-u[q,j]*u[q,i]/u[q,q] to factorize the sparse rows of A * A transpose.
   NOTE: Bound feasibility attained by iteration 1.
   NOTE: Dual feasibility attained by iteration 1.
   NOTE: Constraint feasibility attained by iteration 2.
   NOTE: Primal-Dual Predictor-Corrector Interior point algorithm performed 7 iterations.
   NOTE: Objective = 50075.
   NOTE: The data set WORK.SOLUTION has 18 observations and 14 variables.
   NOTE: There were 18 observations read from the data set WORK.ARCD1.
   NOTE: There were 6 observations read from the data set WORK.MISS_S.
   NOTE: There were 4 observations read from the data set WORK.COND1.
   NOTE: The data set WORK.SOLUTION has 18 observations and 14 variables.

The CONOUT= data set is shown in Figure 5.11.

Oil Industry Example

Crude Oil can come from anywhere

Obs	_from_	_to_	_cost_	_capac_	_lo_	_FLOW_	_FCOST_
1	refinery 1	r1	200	175	50	145.00	29000.00
2	refinery 2	r2	220	100	35	35.00	7700.00
3	r1	ref1 diesel	0	75	0	36.25	0.00
4	r1	ref1 gas	0	140	0	108.75	0.00
5	r2	ref2 diesel	0	75	0	8.75	0.00
6	r2	ref2 gas	0	100	0	26.25	0.00
7	middle east	refinery 1	63	95	20	20.00	1260.00
8	u.s.a.	refinery 1	55	99999999	0	125.00	6875.00
9	middle east	refinery 2	81	80	10	10.00	810.00
10	u.s.a.	refinery 2	49	99999999	0	25.00	1225.00
11	ref1 diesel	servstn1 diesel	18	99999999	0	30.00	540.00
12	ref2 diesel	servstn1 diesel	36	99999999	0	0.00	0.00
13	ref1 gas	servstn1 gas	15	70	0	68.75	1031.25
14	ref2 gas	servstn1 gas	17	35	5	26.25	446.25
15	ref1 diesel	servstn2 diesel	17	99999999	0	6.25	106.25
16	ref2 diesel	servstn2 diesel	23	99999999	0	8.75	201.25
17	ref1 gas	servstn2 gas	22	60	0	40.00	880.00
18	ref2 gas	servstn2 gas	31	99999999	0	0.00	0.00
							50075.00

Figure 5.11: Missing S SUPDEM values in NODEDATA

The optimal supplies of nodes middle east and u.s.a. are 145 and 35 units, respectively. For this example, the same optimal solution is obtained if these nodes had supplies less than these values (each supplies 1 unit, for example) and the THRUNET option was specified in the PROC INTPOINT statement. With the THRUNET option active, when total supply exceeds total demand, the specified nonmissing demand values are the lowest number of flow units that must be absorbed by the corresponding node. This is demonstrated in the following PROC INTPOINT run. The missing S is most useful when nodes are to supply optimal numbers of flow units and it turns out that for some nodes, the optimal supply is zero:

   data miss_s_x;
      missing S;
      input   _node_&$15. _sd_;
      datalines;
   middle east         1
   u.s.a.              1
   servstn1 gas      -95
   servstn1 diesel   -30
   servstn2 gas      -40
   servstn2 diesel   -15
   ;
   proc intpoint
      bytes=100000
      thrunet
      nodedata=miss_s_x     /* No supply (missing S)      */
      arcdata=arcd1         /* the arc descriptions       */
      condata=cond1         /* the side constraints       */
      conout=solution;      /* the solution data set      */
   run;
   proc print;
      var _from_ _to_ _cost_ _capac_ _lo_ _flow_ _fcost_;
      sum _fcost_;
   run;

The following messages appear on the SAS log. Note that the Total supply= 2, not zero as in the last run:

   NOTE: Number of nodes= 14 .
   NOTE: Number of supply nodes= 2 .
   NOTE: Number of demand nodes= 4 .
   NOTE: Total supply= 2 , total demand= 180 .
   NOTE: Number of arcs= 18 .
   NOTE: Number of <= side constraints= 0 .
   NOTE: Number of == side constraints= 2 .
   NOTE: Number of >= side constraints= 2 .
   NOTE: Number of side constraint coefficients= 8 .
   NOTE: The following messages relate to the equivalent Linear Program solved by the Interior Point algorithm.
   NOTE: Number of <= constraints= 0 .
   NOTE: Number of == constraints= 17 .
   NOTE: Number of >= constraints= 2 .
   NOTE: Number of constraint coefficients= 48 .
   NOTE: Number of variables= 20 .
   NOTE: After preprocessing, number of <= constraints= 0.
   NOTE: After preprocessing, number of == constraints= 8.
   NOTE: After preprocessing, number of >= constraints= 2.
   NOTE: The preprocessor eliminated 9 constraints from the problem.
   NOTE: The preprocessor eliminated 20 constraint coefficients from the problem.
   NOTE: After preprocessing, number of variables= 11.
   NOTE: The preprocessor eliminated 9 variables 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 21 nonzero elements in A * A transpose.
   NOTE: Of the 10 rows and columns, 4 are sparse.
   NOTE: There are 15 nonzero superdiagonal elements in the sparse rows of the factored A * A transpose. This includes fill-in.
   NOTE: There are 5 operations of the form u[i,j]=u[i,j]-u[q,j]*u[q,i]/u[q,q] to factorize the sparse rows of A * A transpose.
   NOTE: Bound feasibility attained by iteration 1.
   NOTE: Dual feasibility attained by iteration 1.
   NOTE: Constraint feasibility attained by iteration 2.
   NOTE: Primal-Dual Predictor-Corrector Interior point algorithm performed 7 iterations.
   NOTE: Objective = 50075.
   NOTE: The data set WORK.SOLUTION has 18 observations and 14 variables.
   NOTE: There were 18 observations read from the data set WORK.ARCD1.
   NOTE: There were 6 observations read from the data set WORK.MISS_S_X.
   NOTE: There were 4 observations read from the data set WORK.COND1.
   NOTE: The data set WORK.SOLUTION has 18 observations and 14 variables.

If total supply exceeds total demand, any missing S values are ignored. If total demand exceeds total supply, any missing D values are ignored.

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