Mathematical Formulation of Linear Programming Problem (LPP):
On this page, we discuss the Mathematical Formulation of Linear Programming Problem (LPP) in detail.
тЩж Formulation of LPP:
The procedure of formulation of LPP is as follows:
Step-1:
Identify the decision variables (the variables whose values are to be found out by solving the LPP) and assign the symbols (x1, x2, x3,┬атАж or x, y, z, тАж) to them.
Step-2:
Identify the objective function (to be optimized) and represent it as a linear function of the decision variables.
Z = c1x1 + c2x2 + тАж + cnxn.
Step-3:
Identify all the restrictions or constraints in the problem regarding the resource which are available upto or beyond a certain limit and express them as linear equations and/or inequalities in terms of decision variables in the following manner.
a11x1 + a12x2 + тАж + a1nxn┬а ┬а ┬а(тЙд, =, тЙе)┬а ┬а b1.
Step-4:
Since the negative values of decision variables do not have any valid physical interpretation.
xi┬атЙе 0,┬а ┬а ┬а тИА┬а ┬аi=1, 2, 3, …
тЩж Mathematical Formulation of LPP:
Step-1: Define the decision variables; x1, x2, x3, тАж, xn.
Step-2: Construct the objective function which has to be optimized as a linear equation involving decision variables.
Step-3: Express every condition as a linear inequality involving decision variables.
Step-4: State the non-negativity condition and hence express the given problem as a mathematical model.
тЩж Mathematical Form of LPP:
Let x1, x2, x3, тАж xn be n decision variables. Then the mathematical form of LPP is as follows:
Equation (1) is the objective function, equations (2) are constraints and equation (3) is non-negative restrictions.
In equations (2), (тЙд, =, тЙе) means that any one of three signs may be there.
тЩж Example of Formulation of LPP:
Question:
A person wants to decide the constituents of a diet which will fulfil his daily requirements of proteins, carbohydrates at the minimum cost. The choice is to be made from four different types of foods. The yield per unit of these foods are given in the following table:
Formulate a linear programming model for the problem.
Solution:
Let the number of units of food types 1, 2, 3 and 4 be represented by x1,┬а x2, x3, and x4, respectively.
The cost per unit for food type 1 is Rs. 45, for food type 2 is Rs. 40, for food type 3 is Rs. 85, and for food type 4 is Rs. 65, so the total cost is,
Z = 45 x1 + 40 x2 + 85 x3 + 65 x4.
The objective is to minimize the cost, so the objective function will be,
Minimize Z = 45x1 + 40 x2 + 85 x3 + 65 x4
Since, the number of units of food types cannot be negative, so,
x1 тЙе 0, x2 тЙе 0, x3 тЙе 0, and x4 тЙе 0
Constraints are the fulfilments of daily protein, fat, and carbohydrate requirements. So,
- His daily requirement of proteins is at least 800 units, so by the above table,
3 x1 + 4 x2 + 8 x3 + 6 x4 тЙе 800
- His daily requirement of fats is at least 200 units, so by the above table,
2 x1 + 2 x2 + 7 x3 + 5 x4 тЙе 200
- His daily requirement of carbohydrates is at least 700 units, so by the above table,
6 x1 + 4 x2 + 7 x3 + 4 x4 тЙе 700
Hence, the system of inequalities is,
3 x1 + 4 x2 + 8 x3 + 6 x4 тЙе 800,
2 x1 + 2 x2 + 7 x3 + 5 x4 тЙе 200,
6 x1 + 4 x2 + 7 x3 + 4 x4 тЙе 700.
Hence, the linear programming model (LPP) for this given problem is,
Minimize Z = 45x1 + 40 x2 + 85 x3 + 65 x4
Subject to
3 x1 + 4 x2 + 8 x3 + 6 x4 тЙе 800
2x1 + 2 x2 + 7 x3 + 5 x4 тЙе 200
6x1 + 4 x2 + 7 x3 + 4 x4 тЙе 700
and,
x1, x2, x3, x4 тЙе 0.
(Source тАУ Various books from the college library)
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