1. Formulate Linear Programming Problem (represent mathematically) for Diet Prob
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Question
1. Formulate Linear Programming Problem (represent mathematically) for Diet Problem given In slides. First define decision variables then write objective function and constraints.
The Diet Problem
In the diet model, a list of available foods is given together with
the nutrient content and the cost per unit weight of each food. A
certain amount of each nutrient is required per day. For example,
here is the data corresponding to a civilization with just two types
of grains (G1 and G2) and three types of nutrients (starch,
proteins, vitamins):
The requirement per day of starch, proteins and vitamins is 8, 15
and 3 respectively. The problem is to find how much of each food to
consume per day so as to get the required amount per day of each
nutrient at minimal cost.
G2 5
7 4
2 2
1 0.6
0.35
Explanation / Answer
decision variables are:
• x1: number of units of grain G1 to be consumed per day,
• x2: number of units of grain G2 to be consumed per day
Objective function
the objective is to minimize the total cost per day which is given by
z = 0.6x1 + 0.35x2
Constraints
x1 ? 0 and x2 ? 0.
5x1 + 7x2 ? 8
4x1 + 2x2 ? 15
2x1 + x2 ? 3
This diet problem can therefore be formulated by the following linear program:
Minimize z = 0.6x1 + 0.35x2
subject to:
5x1 + 7x2 ? 8
4x1 + 2x2 ? 15
2x1 + x2 ? 3
x1 ? 0, x2 ? 0.
A solution x = (x1, x2) is said to be feasible with respect to the above linear programif it satisfies all the above constraints. The set of feasible solutions is called the feasible space or feasible region. A feasible solution is optimal if its objective function value is equal to the smallest value z can take over the feasible region
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