PROC OPTLP: The Diet Problem :: SAS/OR(R) 9.22 User's Guide: Mathematical Programming

The OPTLP Procedure

Example 17.3 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 17.3.

Table 17.3 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 set:

data ex3;
input field1 $ field2 $ field3$ field4 field5 $ field6 ;
datalines;
NAME        .          EX3      .     .         .
ROWS        .          .        .     .         .
 N          diet       .        .     .         .
 G          calories   .        .     .         .
 L          protein    .        .     .         . 
 G          fat        .        .     .         .
 G          carbs      .        .     .         .
COLUMNS     .          .        .     .         .
.           br         diet     2     calories  90
.           br         protein  4     fat       1
.           br         carbs    15    .         .
.           mi         diet     3.5   calories  120
.           mi         protein  8     fat       5
.           mi         carbs    11.7  .         .
.           ch         diet     8     calories  106
.           ch         protein  7     fat       9
.           ch         carbs    .4    .         .
.           po         diet     1.5   calories  97
.           po         protein  1.3   fat       .1
.           po         carbs    22.6  .         .
.           fi         diet     11    calories  130
.           fi         protein  8     fat       7
.           fi         carbs    0     .         .
.           yo         diet     1     calories  180
.           yo         protein  9.2   fat       1
.           yo         carbs    17    .         .
RHS         .          .        .     .         .
.           .          calories 300   protein   10
.           .          fat      8     carbs     10
BOUNDS      .          .        .     .         .
UP          .          mi       1     .         .
LO          .          fi       .5    .         .
ENDATA      .          .        .     .         .
;

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

proc optlp data=ex3
  presolver = none 
  solver    = ps 
  primalout = ex3pout 
  dualout   = ex3dout
  printfreq = 1; 
run;

The solution summary and the optimal primal solution are displayed in Output 17.3.1.

Output 17.3.1 Diet Problem: Solution Summary and Optimal Primal Solution

The OPTLP Procedure

Solution Summary

Obs	Label1	cValue1	nValue1
1	Solver	Primal simplex	.
2	Objective Function	diet	.
3	Solution Status	Optimal	.
4	Objective Value	12.081337881	12.081338
5			.
6	Primal Infeasibility	8.881784E-16	8.881784E-16
7	Dual Infeasibility	0	0
8	Bound Infeasibility	0	0
9			.
10	Iterations	5	5.000000
11	Presolve Time	0.00	0
12	Solution Time	0.00	0

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.

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