UNIT 5 Linear Programming Problems F Y B

UNIT 5 Linear Programming Problems F. Y. B. Com Sub- Business Mathematics and Statistics Prof. P. A. Navale Dept. of Commerce

Introduction Ø Linear programming (LP or linear optimisation) is a mathematical method for determining a way to achieve the best outcome (such as maximum profit or lowest cost) in a given mathematical model for some list of requirements represented as linear relationships. Ø Linear programming is a specific case of mathematical programming (mathematical optimisation). More formally, linear programming is a technique for the optimisation of a linear objective function, subject to linear equality and linear inequality constraints. Ø Its feasible region is a convex polyhedron, which is a set defined as the intersection of finitely many half spaces, each of which is defined by a linear inequality. Ø Its objective function is a real-valued affine function defined on this polyhedron.

Definition and Terms of LPP 1) Linear Function : Definition : A function f(x 1, x 2, . . . xn) of x 1, x 2, . . . xn is a linear function if and only if for some set of constants c 1, c 2, . . . . cn f(x 1, x 2, . . . xn) = c 1 x 1 + c 2 x 2 +. . + cnxn Examples: i) x 1 ii) 5 x 1+ 6 x 4 - 2 x 2+1 iii) 3

Definition and Terms of LPP 2)Linear Inequalities : Definition : For any linear functions f(x 1, x 2, . . . xn) and any number b, the inequalities. f(x 1, x 2, . . . xn) £ b and f(x 1, x 2, . . . xn) > b are linear inequalities. Examples : i) x 1 + x 2£ 4 ii) 5 x 1 - 4 > 0 Note : If an inequality can be rewritten as a linear inequality then it is one. Thus x 1 + x 2£ 3 x 3 is a linear inequality because it can be rewritten as x 1+x 2 -3 x 3£ 0. Even x 1 / x 2 £ 4 is a linear inequality because it can be rewritten as x 1 - 4 x 2£ 0. Note that x 1 / x 2 + x 3£ 4 is not a linear inequality, however.

Definition and Terms of LPP 3) Linear Equalities : Definition : For any linear function f(x 1, x 2, . . . xn) and any number b, the equality f(x 1, x 2, . . . xn) = b is a linear equality. 4)LP (Linear Programming) : Definition: A linear Programming problem (LP) is an optimisation problem for which : a)We attempt to maximize (or minimize) a linear function of the decision variables. (objective function) b)The values of the decision variables must satisfy a set of constraints, each of which must be a linear inequality or linear equality. c)A sign restriction on each variable. For each variable xi the sign restriction can either say i) xi> 0, ii) xi£ 0, iii) xi unrestricted (urs).

Definition and Terms of LPP 5) Solution : A Solution to a linear program is a setting of the variables. 6)Feasible region: The feasible region in a linear program is the set of all possible feasible solutions. 7)Feasible solution: A feasible solution to a linear program is a solution that satisfies all constraints. 8)Basic and Non basic variables and solutions : Consider a linear programming problem in which all the constraints are equalities (conversion can be accomplished with slack and excess variables. ) Definitions : 1)If there are n variables and m constraints, a solution with at most m non-zero values is a basic solution. 2)In a basic solution, n-m of the zero-valued variables are considered non-basic variables and the remaining m variables are considered basic variables.

Definition and Terms of LPP 9) Dual of LPP : a)Meaning : Duality is the state of having two distinct but related parts. The concept represents the possibility of an intriguing universal state that has been proposed and debated in diverse disciplines. For example, philosophers speak of mind and matter as underlying all known phenomena; theologians propose the underlying principles of good and bad, and the doctrine that people have both a physical and a spiritual nature; and chemists have proved that every definite compound consists of two parts having opposite electrical activity. It turns out that every LP problem has a related problem called the dual problem or dual. The dual is formulated from information contained in the original problem called the primal problem or primal and when solved, the dual provides all essential information about the solution to the primal.

Formulation of LPP There are mainly four steps in the mathematical formulation of linear programming problem as a mathematical model. Identification of Decision Variables Identification of Set of Constraints Identification of Objective Function Addition of Nonnegativity Restrictions

Formulation of LPP a)Identification of Decision Variables : Identify the decision variables and assign symbols x and y to them. These decision variables are those quantities whose values we wish to determine. b)Identification of Set of Constraints : Identify the set of constraints and express them as linear equations/ inequations in terms of the decision variables. These constraints are the given conditions. c)Identification of Objective Function : Identify the objective function and express it as a linear function of decision variables. It might take the form of maximizing profit or production or minimizing cost. d)Addition of Non-negativity Restrictions : Add the non-negativity restrictions on the decision variables, as in the physical problems, negative values of decision variables have no valid interpretation.

Solution by Graphical Method Linear programming with two decision variables can be solved graphically, Although the method is quite simple the principle of solution is based on certain analytical concepts. This is one of the simplest method to solve Linear Programming Problems : Steps: 1)Treat each inequality into equality. 2)Plot the point and Draw line on Graph for equality. 3)Draw the region whether it is towards the origin i. e. (0, 0) or away from origin shown by ( ) arrow. 4)Find out common feasible region. This may be open region or closed polygon. 5)The optimum value is always at vertices of polygon. 6)Substitute the values of vertices in minimize or maximize objective and find the optimum solution. That is for maximize objective where the solution is maximum and for minimize objective where the solution is minimum

Problems on Linear Programme Ex. No. 1) A company manufactures two products A & B. Both the product passes through two machines. M 1& M 2. The available capacity of each machines time required to each process each unit of product A & B & Profit per unit of product A & B are given below. Product Q 1 A B Available Hours Machine One Machine Two M 1 3 5 4 6 300 Profit per Unit M 2 25 30 320 Formulate the Above Problem as - LPP so as to Maximise profit.

Problems on Linear Programme Solution : I)Decision Variable : The profit is depend upon the No. of products A & B to be produced. Let the No. of products A be produced = X & Let the No. of product B be produce = y. II)Objective Function : Our objectives is to maximize profit. To Maximize profit = f = 25 x + 30 y. . . (I) III)Constraints : M 1 = M 1 requires 3 hours for product A and 4 hours for product B but available Hours are 300. 3 x + 4 y must be 300. . . (II) M 2 = M 2 requires 5 hours for product A and 6 hours for product B but available hours are 320. 5 x + 6 y 320 IV)Formulation of L. P. P : To maximize profit = Subject to = . . . (III) f = 25 x + 30 y 3 x + 4 y 300 5 x + 6 y 320 x> 0 y> 0

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