Introduction to linear programming problem

1. Introduction to linear programming problem

2. Topics to be covered • Introduction to LPP • Definition of LPP • Requirements for LP • Formulation of LP • Example 1 • Example 2

3. Introduction to LPP (Linear programming problem) A large number of military programming and planning problems could be formulated as maximizing/minimizing a linear form of profit/cost function whose variables were restricted to values satisfying a system of linear constraints(set of linear equations/or inequalities). A linear form is a mathematical expression of the type a1+x1+a2x2+…+anxn Where a1,a2,…,an are constants and x1,x2,…,xn are variables. Back

4. Definition of LPP A unique solution for a set of simultaneous equations in n variables(x1,x2,..,xn) at least one of them is nonzero can be if there are exactly n constraints(relations). When the number of relations is greater than or less than n, a unique solutions does not exist but number of solutions can be found. Definition The general LPP calls for optimizing(Max/min)a linear function of variables called the objective function subject to a set of linear equations and/or inequalities called the constraints or restrictions. Back

5. Requirements for LP 1. There must be a well defined objective function which is to be either maximized or minimized and which can expressed as a linear function of decision variable. 2. There must be constraints on the amount of extent of be capable of being expressed as linear equalities in terms of variable. 3. There must be alternative course of action. 4. The decision variable should be inter-related and non-negative. 5. The resource must be limited. Back

6. Formulation of LP First the given problem must be presented in linear programming form. This requires defining the variables of the problem, establishing the relationship between them and formulating the objective function and constraints. If some constraints happen to be nonlinear, they are approximated to appropriate linear functions to fit the linear programming. Back

7. Examples of formulation of LP Example 1:(Production Allocation problem) A firm manufactures two type of products A and B and sells them at a profit of Rs. 2 on type A and Rs. 3 on type B. Each product is processed on two machines G and H. Type A requires one minute of processing time on G and two minutes on H. Type B requires one minute on G and one minute on H. The machine G is available for not more than 6 hour 40 minutes while machine H is available for 10 hours during any working day. Formulate the problem as a Linear Programming problem. Solution: Let x1 be the number of products of type A and x2 the number of products of type B

8. Time of Products (minutes) Available Time (minutes) Machine Type A(X1 Units) Type B(X2 Units) G 1 1 400 H 2 1 600 Profit per Unit Rs. 2 Rs. 3 Since the profit on type A is Rs. 2 per product, 2x1 will be the profit on selling x1 units of type A. Similarly, 3x2 will be the profit on selling x2 units of type B. Therefore, total profit on selling x1 units of A and X2 units of B is given by P=2X1+3X2(Objective Function) Since machine G takes 1 minute time on type A and 1 minute time on type B, the total number of minutes required on machine G is given by: x1+x2

9. Similarly, the total number of minutes required on machine H is given by 2x1+x2 But machine G is not available for more than 6 hours 40 minutes(=400 minutes) X1+X2<=400(First Constraint) Also, the machine H is available for 10 hours only, therefore 2X1+X2<=600(Second Constraint) Since it is not possible to produce negative quantities x1>=0 and x2>=0(non negativity restrictions) Find X1 and X2 such that the profit P=2X1+3X2 is maximum Subject to the conditions: X1+X2<=400, 2X1+X2<=600,X1>=0,X2>=0 Back

10. Example 2: A firm can produce three types of cloth say: A, B and C. Three kinds of wool are required for it, say: red, green and blue wool. One unit length of type A cloth needs 2 meters of red wool and 3 meters of blue wool, One unit length of type B cloth needs 3 meters of red wool, 2 meters of green wool and 2 meters of blue wool and one unit of type C cloth needs 5 meters of green wool and 4 meters of blue wool. The firm has only a stock of 8 meters of red wool, 10 meters of green wool and 15 meters of blue wool. It is assumed that the income obtained from one unit length of type A cloth is Rs. 3.00,of type B cloth is Rs 5.00, and of type C cloth is Rs. 4.00 Determine how the firm should use the available material so as to maximize the income from the finished cloth.

11. Time of Cloth (minutes) Solution: Available Quantity of wool (minutes) Type A(X1 Units) Type B(X2 Units) Quality of wool Type C(X3 Units) Red 2 3 0 8 Green 0 2 5 10 Profit per Unit Rs 3.00 Rs. 5.00 Rs. 4.00 Blue 3 3 4 15

12. Let X1,X2 and X3 be the quantity produced of cloth type A,B,C. Since 2 meters of red wool are required for each meter of cloth A and X1 meters of this type of cloth are produced. So 2X1 meters of red wool will be required for cloth A. Similarly, cloth B requires 3X2 meters of red wool and cloth C does not require red wool. Thus, total quantity of red wool becomes: 2X1+3X2+0X3 Similar for green and blue wool 0X1+2X2+5X3(Green Wool) 3X1+2X2+4X3(Blue Wool)

13. Since not more 8 meters of red, 10 meters of green and 15 meters of blue wools are available, the variables X1,X2,X3 must satisfy the following restrictions:2X1+3X2<=82X2+5X3<=103X1+2X2+4X3<=15Nonnegativity constraints are:X1>=0,X2>=0,X3>=0Total income from the finished cloth is:P=3X1+5X2+4X3 Find X1 and X2 such that the profit P=3X1+5X2 +4X3 is maximum Subject to the conditions: 2X1+3X2<=8,2X2+5X3<=10,3X1+2X2+4X3<=15 X1>=0,X2>=0,X3>=0 Back

14. Thanks