The classic diet-mix problem involves determining the items which should be included in a meal so as
(a) Minimise the cost of the meal while
(b) Satisfying certain nutritional requirements
The nutritional requirements generally take the form of
(a) daily vitamin requirements
(b) restrictions which encourage providing variety in the meals.
(c) restrictions which consider taste and logical similarity of foods etc.
An example would illustrate a simple diet-mix problem
Example. A dietician is planning the menu of a meal to be provided in the hostel of a college. Three main items having different nutritional content have to be served. The dietician wants to ensure at least the minimum daily requirement of each of the three vitamins is provided. The table below gives the details of vitamin content per gram of each type of food, cost per gram of the food and minimum daily requirement for the three vitamins. Any combination of the foods may be provided as long as the total serving is at least 10 grams.
The problem is to determine the number of grams of each food to be included in the meal. The objective is to minimise the cost of each meal subject to satisfying daily minimum requirement of the three vitamins as well as the restriction of minimum serving size.
Food |
Vitamin (mg) |
Cost per gram (Rupees) |
||
1 |
2 |
3 |
||
1 |
50 |
20 |
15 |
0.20 |
2 |
30 |
10 |
60 |
0.40 |
3 |
20 |
30 |
20 |
0.25 |
Minimum daily requirement |
290 |
290 |
250 |
Sol. To formulate the LP model for this problem. Let x_{1},x_{2}and x_{3}be the number of grams in each type of food. The objective function should represent the total cost of the meal in Rupees. The total cost equals the sum of the costs of three items.
Objective function Z = 0.20 x_{1}+ 0.40 x_{2}+ 0.25 x_{3}
Since weare interested in providing at least the minimum daily requirement for each of the three vitamins, there will be three greater than or equal to constraints. The constraint for each type of vitamin will have the form gm from food 1 + gm from food 2 + gm from food 3 Minimum daily requirement.
The constraints are
50 x_{1} + 30 x_{2}+ 20 x_{3≥} 290
20x_{1}+10x_{2}+30x_{3≥} 200
15x_{1}+60x_{2}+20x_{3} ≥250
The restriction that the serving size be at least 10 grams is stated as
x_{1}+x_{2}+x_{3} ≥10
The complete formulation of the problem is as follows:
Minimize Z= 0·20x_{1}+0·40x_{2}+0·25x_{3}
subject to 50 x_{1}+ 30 x_{2}+ 20 x_{3} ≥290
20x_{1}+10x_{2}+30x_{1} ≥200
15x_{1} +60x_{2}+20x_{3} 250
x_{1}+ x_{2}+ x_{1} ≥10
x_{1} , x_{2}, x_{1≥} 0.
Note that non-negativity constraint has been included to ensure that negative quantities of any of the foods will not be recommended.
The above is a very simplified problem involving the planning of one meal, the use of first three food types, and consideration of three vitamins. In actual practice, models have been formulated, which consider
- Menu planning over longer periods of time (daily, weekly, monthly etc)
- The interrelationship of all meals served during a particular day.
- The interrelationship among meals served over the entire planning period
- Many food items and
- Many nutritional requirements.
The number of variables and number of constraints for such models can become extremely large.
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