The OPTLP Procedure

Example 5.3 The Diet Problem

Consider the problem of diet optimization. There are six different foods: bread, milk, cheese, potato, fish, and yogurt. The cost and nutrition values per unit are displayed in Table 3.

Table 3: 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	90	120	106	97	130	180

The objective is to find a minimum-cost diet that contains at least 300 calories, not more than 10 grams of protein, not less than 10 grams of carbohydrates, and not less than 8 grams of fat. In addition, the diet should contain at least 0.5 unit of fish and no more than 1 unit of milk.

You can use the following SAS code to create the MPS-format input data table:

data mylib.ex3;
   input _id_ field1 $ field2 $ field3 $ field4 field5 $ field6;
   datalines;
1  NAME        .          EX3      .     .         .
2  ROWS        .          .        .     .         .
3  N           diet       .        .     .         .
4  G           calories   .        .     .         .
5  L           protein    .        .     .         .
6  G           fat        .        .     .         .
7  G           carbs      .        .     .         .
8  COLUMNS     .          .        .     .         .
9  .           br         diet     2     calories  90
10 .           br         protein  4     fat       1
11 .           br         carbs    15    .         .
12 .           mi         diet     3.5   calories  120
13 .           mi         protein  8     fat       5
14 .           mi         carbs    11.7  .         .
15 .           ch         diet     8     calories  106
16 .           ch         protein  7     fat       9
17 .           ch         carbs    .4    .         .
18 .           po         diet     1.5   calories  97
19 .           po         protein  1.3   fat       .1
20 .           po         carbs    22.6  .         .
21 .           fi         diet     11    calories  130
22 .           fi         protein  8     fat       7
23 .           fi         carbs    0     .         .
24 .           yo         diet     1     calories  180
25 .           yo         protein  9.2   fat       1
26 .           yo         carbs    17    .         .
27 RHS         .          .        .     .         .
28 .           .          calories 300   protein   10
29 .           .          fat      8     carbs     10
30 BOUNDS      .          .        .     .         .
31 UP          .          mi       1     .         .
32 LO          .          fi       .5    .         .
33 ENDATA      .          .        .     .         .
;

You can solve the diet problem by using PROC OPTLP as follows:

proc optlp data=mylib.ex3
   presolver = none
   algorithm = ps
   primalout = mylib.ex3pout
   dualout   = mylib.ex3dout
   logfreq   = 1;
run;

The solution summary and the optimal primal solution are displayed in Output 5.3.1.

Output 5.3.1: Diet Problem: Solution Summary and Optimal Primal Solution

Solution Summary

Obs	Name1	Label1	cValue1	nValue1
1	solver	Solver	LP	.
2	algorithm	Algorithm	Primal Simplex	.
3	objectiveName	Objective Function	diet	.
4	solStatus	Solution Status	Optimal	.
5	objective	Objective Value	12.081337881	12.081338
6				.
7	primalInfeasibility	Primal Infeasibility	0	0
8	dualInfeasibility	Dual Infeasibility	1.44329E-16	1.44329E-16
9	boundInfeasibility	Bound Infeasibility	0	0
10				.
11	iterations	Iterations	6	6.000000
12	presolveTime	Presolve Time	0.00	0.000446
13	solutionTime	Solution Time	0.00	0.001765

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

The cost of the optimal diet is 12.08 units.

Last updated: November 20, 2025