### FORMULATION OF LINEAR PROGRAMMING

Formulation of linear programming is the representation of a problem situation in a mathematical form. It involves well-defined decision variables, with an objective function and set of constraints.

**Objective function**:

The objective of the problem is identified and converted into a suitable objective function. The objective function represents the aim or goal of the system (i.e., decision variables) which has to be determined from the problem. Generally, the objective in most cases will be either to maximize resources or profits or, to minimize the cost or time.

**Constraints**:

When the availability of resources is in surplus, there will be no problem in making decisions. But in real life, organizations normally have scarce resources within which the job has to be performed in the most effective way. Therefore, problem situations are within confined limits in which the optimal solution to the problem must be found.

**Non-negativity constraint**

Negative values of physical quantities are impossible, like producing a negative number of chairs, tables, etc., so it is necessary to include the element of non-negativity as a constraint.

All the characteristics explored above give the following Linear Programming (LP).

A value of (x,y) is in the feasible region if it satisfies all the constraints and signs restrictions. This type of linear programming can be solved by two methods:

1) Graphical method

2) Simplex algorithm method

#### Working Rule of Graphic Method

**Step 1**: Convert the inequality constraint as equations and find the coordinates of the line.

**Step 2**: Plot the lines on the graph. (Note: if the constraint is a type, then the solution zone lies away from the center. If the constraint is s type, then the solution zone is towards the center.)

**Step 3**: Obtain the feasible zone.

**Step 4**: Find the coordinates of the objectives function (profit line) and plot it on the graph representing it with a dotted line.

** Step 5**: Locate the solution point. (Note: If the given problem is maximization, Zmax then locates the solution point at the far most point of the feasible zone from the origin, and if minimization, Zmin then locates the solution at the shortest point of the solution zone from the origin).

**Step 6**: Solution type

i). If the solution point is a single point on the line, take the corresponding values of x and y.

ii). If the solution point lies at the intersection of two equations, then solve for x and y using the two equations.

iii). If the solution appears as a small line, then a multiple solution exists.

iv). If the solution has no confined boundary, the solution is said to be an unbound solution.

### LPP Class 12 Practice Question For State Board

**Question 1.**

Two designers P and Q, earn ₹ 600 and ₹ 800 per day respectively. A can design 12 banners and 8 pairs of posters while B can design 20 banners and 8 pairs of posters per day. To find how many days should each of them work and if it is desired to produce at least 120 banners and 64 posters at a minimum labor cost, formulate this as an LPP.

**Question 2.**

A company produces two types of mobiles, A and B, that require silicon and plastic. Each unit of type A requires 3 g of silicon and 1 g of plastic while that of type B requires 1 g of silicon and 2 g of plastic. The company can use at the most 9 g of silicon and 8 g of plastic. If each unit of type A brings a profit of ₹ 40 and that of type B ₹ 50, find the number of units of each type that the company should produce to maximize profit. Formulate the above LPP and solve it graphically and also find the maximum profit.

**Question 3.**

A man wants to invest an amount of ₹5000. His broker recommends investing in two types of stocks A’ and ‘B’ yielding 30% and 27% return respectively on the invested amount. He decides to invest at least ₹ 2000 in stock A’ and at least ₹ 1000 in stock ‘B’. He also wants to invest at least as much in stock A’ as in stock ‘B’. Solve this linear programming problem graphically to maximise his returns.

**Question 4.**

Find graphically, the maximum value of Z = 2x + 5y,

subject to constraints given below

2x+ 4y ≤ 8;

3x + y ≤ 6;

x + y ≤ A;

x ≥ 0, y ≥ 0.

**Question 5.**

One kind of pizza requires 800 g of flour and 100 g of fat, and another kind of cake requires 400 g of flour and 200 g of fat. Find the maximum number of pizzas which can be made from 20 kg of flour and 4 kg of fat, assuming that there is no shortage of the other ingredients used in making the cakes. Make it an LPP and solve it graphically.