The OPTLP Procedure

Example 12.6 Reoptimizing after Adding a New Constraint

Assume that after solving the diet problem in Example 12.3 you need to add a new constraint on sodium intake of no more than 550 mg/day for adults. The updated nutrition data are given in Table 12.7.

Table 12.7: Updated Cost and Nutrition Values

	Bread	Milk	Cheese	Potato	Fish	Yogurt
Cost	2.0	3.5	8.0	1.5	11.0	1.0
Protein, g	4.0	8.0	7.0	1.3	8.0	9.2
Fat, g	1.0	5.0	9.0	0.1	7.0	1.0
Carbohydrates, g	15.0	11.7	0.4	22.6	0.0	17.0
Calories, Cal	90	120	106	97	130	180
sodium, mg	148	122	337	186	56	132

The input data set ex3 is updated (and the data set is saved as ex6) as follows:

/* added a new constraint to the diet problem */
data ex6;
   input field1 $ field2 $ field3 $ field4 field5 $ field6;
   datalines;
NAME        .          EX6      .     .         .
ROWS        .          .        .     .         .
 N          diet       .        .     .         .
 G          calories   .        .     .         .
 L          protein    .        .     .         .
 G          fat        .        .     .         .
 G          carbs      .        .     .         .
 L          sodium     .        .     .         .
COLUMNS     .          .        .     .         .
.           br         diet     2     calories  90
.           br         protein  4     fat       1
.           br         carbs    15    sodium    148
.           mi         diet     3.5   calories  120
.           mi         protein  8     fat       5
.           mi         carbs    11.7  sodium    122
.           ch         diet     8     calories  106
.           ch         protein  7     fat       9
.           ch         carbs    .4    sodium    337
.           po         diet     1.5   calories  97
.           po         protein  1.3   fat       .1
.           po         carbs    22.6  sodium    186
.           fi         diet     11    calories  130
.           fi         protein  8     fat       7
.           fi         carbs    0     sodium    56
.           yo         diet     1     calories  180
.           yo         protein  9.2   fat       1
.           yo         carbs    17    sodium    132
RHS         .          .        .     .         .
.           .          calories 300   protein   10
.           .          fat      8     carbs     10
.           .          sodium   550   .         .
BOUNDS      .          .        .     .         .
UP          .          mi       1     .         .
LO          .          fi       .5    .         .
ENDATA      .          .        .     .         .
;

For the modified problem you can warm start the primal and dual simplex algorithms to get a solution faster. The dual simplex algorithm is preferred because a dual feasible solution can be readily constructed from the optimal solution to the diet optimization problem.

Since there is a new constraint in the modified problem, you can use the following SAS code to create a new DUALIN= data set ex6din with this information:

data ex6newcon;
   _ROW_='sodium  '; _STATUS_='A';
   output;
run;

/* create a new DUALIN= data set to include the new constraint */
data ex6din;
   set ex3dout ex6newcon;
run;

Note that this step is optional. In this example, you can still use the data set ex3dout as the DUALIN= data set to solve the modified LP problem by using the BASIS=WARMSTART option. PROC OPTLP validates the PRIMALIN= and DUALIN= data sets against the input model. Any new variable (or constraint) in the model is added to the PRIMALIN= (or DUALIN=) data set, and its status is assigned to be 'A'. The primal and dual simplex algorithms decide its corresponding status internally. Any variable in the PRIMALIN= and DUALIN= data sets but not in the input model is removed.

The _ROW_ and _STATUS_ columns of the DUALIN= data set ex6din are shown in Output 12.6.1.

Output 12.6.1: DUALIN= Data Set with a Newly Added Constraint

Obs	_ROW_	_STATUS_
1	calories	U
2	protein	L
3	fat	U
4	carbs	B
5	sodium	A

The dual simplex algorithm is called to solve the modified diet optimization problem more quickly with the following SAS code:

proc optlp data=ex6
   objsense=min
   presolver=none
   algorithm=ds
   primalout=ex6pout
   dualout=ex6dout
   scale=none
   logfreq=1
   basis=warmstart
   primalin=ex3pout
   dualin=ex6din;
run;

The optimal primal and dual solutions of the modified problem are displayed in Output 12.6.2.

Output 12.6.2: Primal and Dual Solution Output

Primal Solution

Obs	Objective Function ID	Variable Name	Variable Type	Objective Coefficient	Lower Bound	Upper Bound	Variable Value	Variable Status	Reduced Cost
1	diet	br	N	2.0	0.0	1.7977E308	0.00000	L	1.19066
2	diet	mi	D	3.5	0.0	1	0.05360	B	0.00000
3	diet	ch	N	8.0	0.0	1.7977E308	0.44950	B	0.00000
4	diet	po	N	1.5	0.0	1.7977E308	1.86517	B	0.00000
5	diet	fi	O	11.0	0.5	1.7977E308	0.50000	L	5.15641
6	diet	yo	N	1.0	0.0	1.7977E308	0.00000	L	1.10849

Dual Solution

Obs	Objective Function ID	Constraint Name	Constraint Type	Constraint RHS	Constraint Lower Bound	Constraint Upper Bound	Dual Variable Value	Constraint Status	Constraint Activity
1	diet	calories	G	300	.	.	0.02179	U	300.000
2	diet	protein	L	10	.	.	-0.55360	L	10.000
3	diet	fat	G	8	.	.	1.06286	U	8.000
4	diet	carbs	G	10	.	.	0.00000	B	42.960
5	diet	sodium	L	550	.	.	0.00000	B	532.941

The iteration log shown in Output 12.6.3 indicates that it takes the dual simplex algorithm no more iterations to solve the modified problem by using the BASIS=WARMSTART option, since the optimal solution to the original problem remains optimal after one more constraint is added.

Output 12.6.3: Iteration Log

NOTE: The OPTLP procedure is executing in single-machine mode.

NOTE: The problem EX6 has 6 variables (0 free, 0 fixed).

NOTE: The problem has 5 constraints (2 LE, 0 EQ, 3 GE, 0 range).

NOTE: The problem has 29 constraint coefficients.

NOTE: The LP presolver value NONE is applied.

NOTE: The LP solver is called.

NOTE: The Dual Simplex algorithm is used.

Objective Entering Leaving

Phase Iteration Value Time Variable Variable

D 2 1 1.208134E+01 0

NOTE: Optimal.

NOTE: Objective = 12.081337881.

NOTE: The Dual Simplex solve time is 0.00 seconds.

NOTE: The data set WORK.EX6POUT has 6 observations and 10 variables.

NOTE: The data set WORK.EX6DOUT has 5 observations and 10 variables.

Both this example and Example 12.4 illustrate the situation in which the optimal solution does not change after some perturbation of the parameters of the LP problem. The simplex algorithm starts from an optimal solution and quickly verifies the optimality. Usually the optimal solution of the slightly perturbed problem can be obtained after performing relatively small number of iterations if starting with the optimal solution of the original problem. In such cases you can expect a dramatic reduction of computation time, for instance, if you want to solve a large LP problem and a slightly perturbed version of this problem by using the BASIS=WARMSTART option rather than solving both problems from scratch.