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A Linear Programming DASH diet model for persons with hypertension has previously been formulated and daily minimum cost diet plans that satisfy the DASH diets’ tolerable intake level of the nutrients for 1500 mg a day Sodium level and different daily calorie levels were obtained using sample foods from the DASH diet eating plan chart. But the limitation in the use of linear programming model in selecting diet plans to meet specific nutritional requirements which normally results in the oversupply of certain nutrients was evident in the linear programming DASH diet plan obtained as the nutrient level of the diet plans obtained had wide deviations of from the DASH diets’ tolerable upper and lower intake level for the given calorie and sodium levels. Hence the need for a model that gives diet plans with minimized nutrients’ level deviations from the DASH diets’ tolerable intake level for different daily calorie and sodium level at desired cost. A weighted Goal Programming DASH diet model that minimizes the daily cost of the DASH eating plan as well as deviations of the diets’ nutrients content from the DASH diet’s tolerable intake levels is hereby presented in this work. The formulated weighted goal programming DASH diet model is further illustrated using chosen sample foods from the DASH food chart as used in the work on the linear programming DASH diet model for a 1500 mg sodium level and 2000 calories a day diet plan as well as for 1800, 2200, 2400, 2600, 2800 and 3000 daily calorie levels. A comparison of the DASH nutrients’ composition of the weighted Goal Programming DASH diet plans and those of the linear programming DASH diet plans were carried out at this sodium level and the different daily calorie levels. It was evident from the results of the comparison that the weighted goal programming DASH diet plans has minimized deviations from the DASH diet’s tolerable intake levels than those of the linear programming DASH diet plans.

The DASH eating plan has been shown by research to prevent or lower high blood pressure. The DASH heart healthy daily eating plan requires foods that have low sodium, saturated fat, total fat and cholesterol nutrient content while rich in potassium, magnesium, calcium and fiber: see [

The goal programming technique has been used by many researchers to model diet problems. [

The linear programming DASH diet model is given as follows:

Minimize D C = C 1 X 1 + C 2 X 2 + C 3 X 3 + ⋯ + C n X n

Subject to the Constraints

a 11 X 1 + a 12 X 2 + a 13 X 3 + ⋯ + a 1 n X n ≤ R 1 c

(Constraint on total fat)

a 21 X 1 + a 22 X 2 + a 23 X 3 + ⋯ + a 2 n X n ≤ R 2 c

(Constraint on sodium)

a 31 X 1 + a 32 X 2 + a 33 X 3 + ⋯ + a 3 n X n ≤ R 3 c

(Constraint on cholesterol)

a 41 X 1 + a 42 X 2 + a 43 X 3 + ⋯ + a 4 n X n ≤ R 4 c

(Constraint on saturated fat)

a 51 X 1 + a 52 X 2 + a 53 X 3 + ⋯ + a 5 n X n ≥ R 5 c

(Constraint on calcium)

a 61 X 1 + a 62 X 2 + a 63 X 3 + ⋯ + a 6 n X n ≥ R 6 c

(Constraint on magnesium)

a 71 X 1 + a 72 X 2 + a 73 X 3 + ⋯ + a 7 n X n ≥ R 7 c

(Constraint on fiber)

a 81 X 1 + a 82 X 2 + a 83 X 3 + ⋯ + a 8 n X n ≥ R 8 c

(Constraint on potassium)

a 91 X 1 + a 92 X 2 + a 93 X 3 + ⋯ + a 9 n X n = R c

(Constraint on calorie)

X j ≥ S L j c

X j ≤ S H j c

where X 1 , X 2 , X 3 , ⋯ , X n ≥ 0 ,

j = 1 , 2 , 3 , ⋯ , n , where j is number of food items;

i = 1 , 2 , 3 , ⋯ , 9 , where i is number of nutrients.

where DC = Diet Cost,

a 11 , a 12 , a 13 , ⋯ , a 8 n are the content of the “DASH nutrients” in the foods.

R 1 c , R 2 c , R 3 c and R 4 c are the Tolerable Upper Intake Level for Total fat, Sodium Cholesterol and Saturated fat respectively for calorie level c.

R 5 c , R 6 c , R 7 c and R 8 c are the Tolerable Lower Intake Level for Calcium, Magnesium, Fiber and Potassium respectively for calorie level c.

R c is the Calorie level.

S L j c is the estimated Least number of daily Servings of food item j for calorie level c.

S H j c is the estimated Highest number of daily Servings of food item j for calorie level c.

The Weighted Goal programming model is used when the decision maker is interested in minimizing the deviations of set goals of relatively equal importance and assigns weights as coefficients to the deviational variables in the objective function to better reflect their importance and desirability of the deviations from the various goals. [

Minimize Z = ∑ i = 1 m ( w i + d i + + w i − d i − )

Subject to the constraints

Goal constraints: ∑ j = 1 n a i j x j + d i − − d i + = b i for i = 1 , ⋯ , m

System constraints: ∑ j = 1 n a i j x j = b i

With d i + , d i − , x j ≥ 0 , for i = 1 , ⋯ , m and j = 1 , ⋯ , n

where

Z is the objective function,

d i − and d i + are the negative and positive deviational variables (underachievement and overachievement) in goal i,

w i − and w i + are non-negative constants representing the relative weights to be assigned to the respective positive and negative deviation variables,

a i j is the coefficient associated with variable j in goal i.

With i = 1 , ⋯ , m and j = 1 , ⋯ , n .

The DASH eating plan is rich in fruits, vegetables, fat free or low-fat milk and milk products, whole grains, fish, poultry, beans, seeds and nuts. It also contains less salt and sodium, sweets, added sugar, and sugar-containing beverages, fats and red meats than most typical diets. The DASH eating plan for each individual according to the “Guide to lowering your Blood Pressure with DASH” depends on the number of calories that individual is allowed each day. The calorie levels which includes 1800, 2000, 2200, 2400, 2600, 2800 and 3000 calories a day depends on the age and activity level of the individual. Different calorie needs for each of the activity levels and for different ages for both males and females is shown in the appendices. The DASH eating plan is also based on recommended levels of daily sodium intake. This includes the 2300 and 1500 milligrams Sodium levels that DASH allows each day.

The decision variables for the weighted Goal Programming DASH diet model are x 1 , x 2 , x 3 , ⋯ , x n which represents the daily number of servings of foods 1 , 2 , 3 , ⋯ , n in the diet plan.

The target goals to be achieved include:

- Goal 1 (Cost goal): Minimize the overachievement of the daily target cost of the diet, C g .

- Goal 2 (Sodium nutrient goal): Minimize the overachievement of the Tolerable Intake Level of sodium ( S g ), in milligrams (mg).

- Goal 3 (Cholesterol goal): Minimize the overachievement of the Tolerable Intake Level of cholesterol, ( C O L m g ), in milligrams (mg).

- Goal 4 (Total fat goal): Minimize the overachievement of the Tolerable Intake Level of total fat ( T F g ), in grams (g).

- Goal 5 (Calorie goal): Attain the allowed daily calorie level ( C A L O g ).

- Goal 6 (Saturated fat goal): Minimize the overachievement of the Tolerable Intake Level of saturated fat ( S F m g ), in milligrams (mg).

- Goal 7 (Magnesium goal): Minimize the underachievement of the Tolerable Intake Level of magnesium ( M m g ), in milligrams (mg).

- Goal 8 (Fiber goal): Minimize the underachievement of the Tolerable Intake Level of fiber ( F I B g ), in grams (g).

- Goal 9 (Potassium goal): Minimize the underachievement of the Tolerable Intake Level of potassium, P m g , in milligrams (mg).

- Goal 10 (Calcium goal): Minimize the underachievement of the Tolerable Intake Level of calcium, C A L m g , in milligrams (mg).

Because we don’t have the same unit of measurements for the different goals, hence the variables in the objective function will represent percentage deviation rather than absolute deviation.

So the objective function that minimizes the sum of the percentage deviations from the targets is given as;

Minimize D = w C + d C + b C ⋅ 100 1 + w S + d S + b S ⋅ 100 1 + w c o l + d c o l + b c o l ⋅ 100 1 + w T F + d T F + b T F ⋅ 100 1 + w S F + d S F + b S F ⋅ 100 1 + w M − d M − b M ⋅ 100 1 + w F i b − d F i b − b F i b ⋅ 100 1 + w P − d P − b P ⋅ 100 1 + w C a l − d C a l − b C a l ⋅ 100 1 + ( w c a l o − d c a l o − b c a l o + w c a l o + d c a l o + b c a l o ) ⋅ 100 1

Subject to the constraints

c 1 x 1 + c 2 x 2 + c 3 x 3 + ⋯ + c n x n + ( 0 ) d c − + ( 1 ) d c + = C ( $ )

(Cost goal constraint in Dollars)

a 11 x 1 + a 12 x 2 + a 13 x 3 + ⋯ + a 1 n x n + ( 0 ) d s − − ( 1 ) d s + = S ( g )

(Sodium goal constraint in grams)

a 21 x 1 + a 22 x 2 + a 23 x 3 + ⋯ + a 2 n x n + ( 0 ) d c o l − − ( 1 ) d c o l + = C O L ( m g )

(Cholesterol goal constraint in milligrams)

a 31 x 1 + a 32 x 2 + a 33 x 13 + ⋯ + a 3 n x n + ( 0 ) d t f − − ( 1 ) d t f + = T F ( g )

(Total fat goal constraint in grams)

a 41 x 1 + a 42 x 2 + a 43 x 3 + ⋯ + a 4 n x n + ( 0 ) d s f − − ( 1 ) d s f + = S F ( m g )

(Saturated fat goal constraint in milligrams)

a 51 x 1 + a 52 x 2 + a 53 x 3 + ⋯ + a 5 n x n + ( 1 ) d M − − ( 0 ) d M + = M ( m g )

(Magnesium goal constraint in milligrams)

a 61 x 1 + a 62 x 2 + a 63 x 3 + ⋯ + a 6 n x n + ( 1 ) d f i b − − ( 0 ) d f i b + = F I B ( g )

(Fiber goal constraint in grams)

a 71 x 1 + a 72 x 2 + a 73 x 3 + ⋯ + a 7 n x n + ( 1 ) d P − − ( 0 ) d P + = P ( m g )

(Potassium goal constraint in milligrams)

a 81 x 1 + a 82 x 2 + a 83 x 3 + ⋯ + a 8 n x n + ( 1 ) d C a − − ( 0 ) d C a + = C A ( m g )

(Calcium goal constraint in milligrams)

a 91 x 1 + a 92 x 2 + a 93 x 3 + ⋯ + a 9 n x m + ( 0 ) d c a l o − + ( 1 ) d c a l o + = C A L O

(Calorie goal Constraint)

With d C + , d S + , d c o l + , d T F + , d S F + , d M − , d F i b − , d P − , d C a l − and d c a l o + , d c a l o − ≥ 0

where:

d C + , d S + , d c o l + , d t f + , d s f + , d M + , d f i b + , d P + , d C a + , d c a l o + and d C − , d S − , d c o l − , d t f − , d s f − , d M − , d f i b − , d P − , d C a − , d c a l o − ≥ 0 represents the overachievements and under achievements respectively of the cost, sodium, cholesterol, total fat, saturated fat, magnesium, fiber, potassium, calcium and calorie level,

w C + , w S + , w C o l + , w T F + , w S F + , w M − , w F i b − , w P − , w C a l − , w C a l o + and w C a l o + are the weights assigned to the respective overachievements and under achievements of the diet cost, DASH nutrients and calorie level.

Weights are assigned to indicate desirable and undesirable deviations. Weights of “1” are assigned to deviations that are undesirable in the model while weights of “0” are assigned to deviations that are desirable in the model [

b C , b S , b C o l , b T F , b M , b F i b , b P , b C a l and b C a l o are the target values of the diet cost, DASH nutrients and calorie level respectively.

a i j = Quantity of ith DASH nutrient in jth food (in g or mg depending on the nutrient).

with i = 1 , 2 , ⋯ , 8 and j = 1 , 2 , ⋯ , n .

a 9 j = Calorie level of jth food (in calorie)

C j , cost of 1 serving of jth food.

x j = is the number of servings of food j in the diet plan.

C g is the budgeted cost of the eating plan.

S g , C O L g , T F g and S F g are the Tolerable target level of Sodium, Cholesterol, Total Fat and Saturated Fat respectively in the days’ diet plan while M g , F I B g , P 0 g and C A L C g are the tolerable target level of magnesium, fiber, potassium and calcium intake level respectively in the days’ diet plan.

The formulated weighted goal programming DASH diet model will be illustrated using the same data of selected food items from the DASH sample food chat and their cost per serving as used in work by [

The first part of

DASH diet model and that of the LP DASH diet model with respect to the DASH nutrients tolerable intake level for 1500 mg sodium level for 1800, 2200, 2400, 2600, 2800 and 3000 calorie levels. As in the case of the comparison of the 2000 daily calorie level we see that the Weighted Goal programming model gives eating plans with minimized nutrient deviations from the targeted DASH nutrients’ tolerable target levels while in the cases of the Linear programming model there are very large nutrient deviations from targeted DASH nutrients’ tolerable intake levels.

The Weighted Goal Programming DASH diet model that minimizes the daily cost of the DASH eating plan as well as deviations of the daily diet plans’ nutrient content from the targeted DASH diet’s tolerable intake levels has been formulated. It was further illustrated using the chosen sample foods from the DASH food chart for 1500 mg sodium level and 2000 calories a day diet plan and then also for 1800, 2200, 2400, 2600, 2800 and 3000 calorie levels. The “DASH nutrients” composition between the Weighted Goal Programming DASH diet model and the Linear Programming DASH diet model eating plans for different daily calorie levels was compared. From the comparison of the percentage deviation of the nutrient contents of the eating plan of two models from the DASH nutrients’ tolerable target level for 2000 calorie and 1500 mg sodium daily levels as shown in

Iwuji, A.C. and Agwu, E.U. (2017) A Weighted Goal Programming Model for the DASH Diet Problem: Comparison with the Linear Programming DASH Diet Model. American Journal of Operations Research, 7, 307-322. https://doi.org/10.4236/ajor.2017.75023

1) Formulated weighted goal programming model for 2000 calorie a day eating plan with 1500 mg sodium level

Minimize D = ( 1 ) d C + 944 ⋅ 100 1 + ( 1 ) d S + 1500 ⋅ 100 1 + ( 1 ) d c o l + 129 ⋅ 100 1 + ( 1 ) d T F + 68 ⋅ 100 1 + ( 1 ) d S F + 16 ⋅ 100 1 + ( 1 ) d M − 542 ⋅ 100 1 + ( 1 ) d F i b − 34 ⋅ 100 1 + ( 1 ) d P − 4721 ⋅ 100 1 + ( 1 ) d C a l − 1334 ⋅ 100 1 + ( ( 1 ) d c a l o − 2000 + ( 1 ) d c a l o + 2000 ) ⋅ 100 1

Subject to the constraints

15 x 1 + 20 x 2 + 15 x 3 + 15 x 4 + 30 x 5 + 15 x 6 + 15 x 7 + 50 x 8 + ( 0 ) d c − + ( 1 ) d c + = 944 Naira

(Cost goal constraint in Naira)

33.6 x 1 + 1.5 x 2 + 124.8 x 3 + 15 x 4 + 8.1 x 5 + 3.2 x 6 + 2.4 x 7 + 73 x 8 + ( 0 ) d s − − ( 1 ) d s + = 1500 mg

(Sodium goal constraint in grams)

3 x 5 + 0.29 x 8 + ( 0 ) d c o l − − ( 1 ) d c o l + = 129 mg

(Cholesterol goal constraint in milligrams)

0.24 x 1 + 11.48 x 2 + 0.58 x 3 + 0.30 x 4 + 0.10 x 5 + 0.48 x 6 + 0.16 x 7 + 4.1 x 8 + ( 0 ) d f − − ( 1 ) d f + = 68 g

(Total fat goal constraint in grams)

0 x 1 + 1.55 x 2 + 0.2 x 3 + 0 x 4 + 0.6 x 5 + 0 x 6 + 0 x 7 + 34 x 8 + ( 0 ) d s f − − ( 1 ) d s f + = 16 g

(Saturated fat goal constraint in milligrams)

9.6 x 1 + 47.75 x 2 + 13.25 x 3 + 14 x 4 + 2.4 x 5 + 17.6 x 6 + 8 x 7 + 43 x 8 + ( 1 ) d m a g − − ( 0 ) d m a g + = 542 mg

(Magnesium goal constraint in milligrams)

2.48 x 1 + 2.33 x 2 + 1.55 x 3 + 3 x 4 + 0 x 5 + 2.72 x 6 + 0.24 x 7 + 0 x 8 + ( 1 ) d f i b − − ( 0 ) d f i b + = 34 g

(Fiber goal constraint in grams)

212.8 x 1 + 181.75 x 2 + 56.5 x 3 + 264 x 4 + 31 x 5 + 265.6 x 6 + 87.2 x 7 + 397 x 8 + ( 1 ) d P − − ( 0 ) d P + = 4721 mg

(Potassium goal constraint in milligrams)

28 x 1 + 4.25 x 2 + 12.25 x 3 + 24 x 4 + 25 x 5 + 49.6 x 6 + 5.6 x 7 + 40 x 8 + ( 1 ) d C a l − − ( 0 ) d C a l + = 1334 mg

(Calcium goal constraint in milligrams)

28 x 1 + 144.5 x 2 + 58.5 x 3 + 90 x 4 + 7 x 5 + 72 x 6 + 23.2 x 7 + 151 x m + ( 0 ) d c a l o − + ( 1 ) d c a l o + = 2000 calories

(Calorie goal Constraint).

2) Linear Programming DASH diet model for 2000 calorie and 1500 mg sodium levels a day eating plan.

Minimize D C = 15 X 1 + 20 X 2 + 15 X 3 + 15 X 4 + 30 X 5 + 15 X 6 + 15 X 7 + 50 X 8

Subject to the constraints

0.24 X 1 + 11.48 X 2 + 0.58 X 3 + 0.30 X 4 + 0.10 X 5 + 0.48 X 6 + 0.16 X 7 + 4.1 X 8 ≤ 68

(Constraint on total fat)

33.60 X 1 + 1.5 X 2 + 124.8 X 3 + 15 X 4 + 8.1 X 5 + 3.2 X 6 + 2.4 X 7 + 73 X 8 ≤ 1500

(Constraint on sodium)

0 X 1 + 0 X 2 + 0 X 3 + 0 X 4 + 3 X 5 + 0 X 6 + 0 X 7 + 0.29 X 8 ≤ 129

(Constraint on cholesterol)

0 X 1 + 1.55 X 2 + 0.2 X 3 + 0 X 4 + 0.6 X 5 + 0 X 6 + 0 X 7 + 340 X 8 ≤ 16

(Constraint on saturated fat)

28 X 1 + 4.25 X 2 + 12.25 X 3 + 24 X 4 + 250 X 5 + 49.6 X 6 + 5.6 X 7 + 40 X 8 ≥ 1334

(Constraint on calcium)

9.6 X 1 + 47.75 X 2 + 13.25 X 3 + 14 X 4 + 2.4 X 5 + 17.6 X 6 + 8 X 7 + 43 X 8 ≥ 542

(Constraint on magnesium)

2.48 X 1 + 2.33 X 2 + 1.55 X 3 + 3 X 4 + 0 X 5 + 2.72 X 6 + 0.24 X 7 + 0 X 8 ≥ 34

(Constraint on fiber)

212.8 X 1 + 181.75 X 2 + 56.5 X 3 + 264 X 4 + 31 X 5 + 262.6 X 6 + 87.2 X 7 + 397 X 8 ≥ 4721

(Constraint on potassium)

20 X 1 + 144.5 X 2 + 58.5 X 3 + 90 X 4 + 7 X 5 + 72 X 6 + 23.2 X 7 + 151 X 8 = 2000

(Constraint on calorie)

X 1 ≥ 4

X 2 ≤ 1

X 3 ≥ 3

X 4 ≥ 4

X 5 ≥ 6

X 6 ≥ 4

X 7 ≥ 4

X 8 ≤ 6

X 1 ≤ 20

X 3 ≤ 8

X 4 ≤ 6

X 5 ≤ 9

X 6 ≤ 8

X 7 ≤ 9

3)

4) DASH daily calorie need chart for different level of activities

5) Tolerable target intake level of DASH nutrients for 1500 milligrams sodium level and 2000 calories a day DASH eating plan