Mathematical Formulation of Linear Programming Problem (LPP):

On this page, we discuss the Mathematical Formulation of Linear Programming Problem (LPP) in detail.

♦ 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, … or x, y, z, …) 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 + … + c_nx_n.

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 + … + a_1nx_n     (≤, =, ≥)    b₁.

Step-4:

Since the negative values of decision variables do not have any valid physical interpretation.

x_i ≥ 0,      ∀   i=1, 2, 3, …

♦ Mathematical Formulation of LPP:

Step-1: Define the decision variables; x₁, x₂, x₃, …, x_n.

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.

♦ Mathematical Form of LPP:

Let x₁, x₂, x₃, … x_n 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), (≤, =, ≥) means that any one of three signs may be there.

♦ 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,  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 ≥ 0, x₂ ≥ 0, x₃ ≥ 0, and x₄ ≥ 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 ≥ 800

• His daily requirement of fats is at least 200 units, so by the above table,

2 x₁ + 2 x₂ + 7 x₃ + 5 x₄ ≥ 200

• His daily requirement of carbohydrates is at least 700 units, so by the above table,

6 x₁ + 4 x₂ + 7 x₃ + 4 x₄ ≥ 700

Hence, the system of inequalities is,

3 x₁ + 4 x₂ + 8 x₃ + 6 x₄ ≥ 800,

2 x₁ + 2 x₂ + 7 x₃ + 5 x₄ ≥ 200,

6 x₁ + 4 x₂ + 7 x₃ + 4 x₄ ≥ 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 ≥ 800

2x₁ + 2 x₂ + 7 x₃ + 5 x₄ ≥ 200

6x₁ + 4 x₂ + 7 x₃ + 4 x₄ ≥ 700

and,

x₁, x₂, x₃, x₄ ≥ 0.

(Source – Various books from the college library)

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