# Formulation of Linear Programming-Minimization Case

**Definition:** **Linear programming** is a technique for selecting the best alternative from the set of available alternatives, in situations in which the objective function and constraint function can be expressed in quantitative terms.

**Minimization Case: **The minimization case can be well understood through a problem. Let’s say; the agricultural research institute recommended a farmer to spread out at least 5000 kg of phosphate fertilizer and not less than 7000 kg of nitrogen fertilizer to raise the productivity of his crops on the farm. There are two mixtures A and B, weighs 100 kg each, from which these fertilizers can be obtained.

The cost of each Mixture A and B is Rs 40 and 25 respectively. Mixture A contains 40 kg of phosphate and 60 kg of nitrogen while the Mixture B contains 60 kg of phosphate and 40 kg of nitrogen. This problem can be represented as a linear programming problem to find out how many bags of each type a farmer should buy to get the desired amount of fertilizers at the minimum cost.

Firstly, the objective function is to be formulated. Suppose, x_{1 }and x_{2} are the number of bags of mixture A and mixture B. The cost of both the mixture is 40_{x1 }+ 25_{x2} and thus, the objective function will be:

**Minimize**

**Z = 40x _{1}+25x_{2}**

In this problem, there are two constraints, minimum 5000 kg of phosphate and minimum 7000 kg of nitrogen is required. The Bag A contains 40 kg of phosphate while Bag B contains 60 kg of phosphate. Thus, the phosphate constraint can be expressed as:

**40x _{1 }+ 60x_{2 }**

**≥**

**5000**

Similarly, the second constraint equation can be expressed as:

**60x _{1 }+ 40x_{2 }**

**≥**

**7000**

Where, Bag A contains 60 kg of nitrogen and Bag B contains 40 kg of nitrogen, and the minimum requirement of nitrogen is 7000 kg.

Thus, the linear programming problem is:

Minimize** Z = 40x _{1}+25x_{2} (cost)**

Subject to:

**40x _{1 }+ 60x_{2 }**

**≥**

**5000 (Phosphate Constraint)**

60x

60x

_{1 }+ 40x_{2 }**≥**

**7000 (Nitrogen Constraint)**

x

x

_{1}, x_{2 }**≥**

**0 (Non-negativity Restriction)**

**Note: **It is to be noted that**, “****≥****” (greater than equal to) **sign shows the full utilization of resources at the minimum cost. The non-negativity condition is used, since x_{1} and x_{2} represent the number of bags of both the mixture and hence cannot have the negative values.