LINEAR PROGRAMMING

LINEAR PROGRAMMING

Linear Programming • Technique for economic allocation of scarce resources like man, machine, material etc to several competing activities such as products, services etc on the basis of given criterion of optimality. • Linear relationships used in mathematical modeling LINEAR PROGRAMMING

General Structure of LP • LP model, as in any OR model, includes three basic elements: • Decision variables that we seek to determine • Objective (goal) that we aim to achieve • Constraints that we need to satisfy • Maximize or Minimize a Linear objective subject to linear equalities or non-equalities (constraints). LINEAR PROGRAMMING

Terminology • Objective Function - Value measure used to rank alternatives, Seek to maximize or minimize this objective, examples: maximize NPV, minimize cost. Example 3x + 5y , x & y are decision variables. • Decision Variables - Quantities you can control to improve your objective which should completely describe the set of decisions to be made. • Constraints - Limitations on the values of the decision variables. Example 2x + 3y ≤24, x ≥0, y ≥0 LINEAR PROGRAMMING

Mathematical Model Optimize (max or min) Z Z = measure of performance variable c1, c2, …cn represent contribution of a unit of the respective variable to performance of z x1, x2, …. xn are decision variables subject to linear , non-negative constraints. LINEAR PROGRAMMING

Algebraic Formulation Optimize (max or min) the objective function subject to linear constraints i = 1,2,…m constraints non-negativity constraints LINEAR PROGRAMMING

Guidelines on LP model Formulation • Step 1 – Identify decision variables. • Express each constraint in words. Identify the equality condition • Express objective function verbally • Identify the decision variables verbally • Step 2 – Identify the problem data • Step 3 – Formulate the constraints • Convert the verbal constraint equation in mathematical form. • Step 4 – Formulate the objective function in mathematical form LINEAR PROGRAMMING

Example of LP Model • Reddy Mikks produces both interior and exterior paints from two raw materials, M1 and M2. The following table provides the basic data of the problem A market survey restricts the maximum demand of interior paint to 2 tons. Additionally, the daily demand of interior paint cannot exceed that of exterior paint by more than 1 ton. Reddy Mikks wants to determine the optimum (best) product mix of interior and exterior paints that maximize daily profit. LINEAR PROGRAMMING

Reddy Mikks – Objective Function • We need to determine the amount of interior and exterior paints to be produced. • The decision variables in this problem are defined as • x1 – Exterior paint produced per day • x2 – Interior paint produced per day • The objective is to maximize profit. Contribution of exterior is $5000/Ton and interior paint is $4000/Ton. • Objective function will be • Maximize Z (profit) = 5x1 + 4x2 LINEAR PROGRAMMING

Reddy Mikks - Constraints • The first set of constraints is the raw material • Usage of raw material by both paints must be less than available raw materials. • Usage of raw material M1 is 6x1+4x2 for exterior and interior paints. • Similarly for M2 it is 1x1+2x2 for exterior and interior paints. • Available raw material is 24 Tons of M1 and 6 Tons of M2 • Hence constraint equations are • 6x1 + 4x224 (Raw material M1) • x1 + 2x26 (Raw material M2) LINEAR PROGRAMMING

Reddy Mikks - Constraints • The second set of constraints is the demand. • Maximum daily demand of interior paint is 2 T • And the demand for interior paint over exterior paint cannot exceed 1 T. • Expressed mathematically these are • x2 2 • x2 – x1  1 • Finally the non-negativity constraint gives us • x1, x2 0 LINEAR PROGRAMMING

The Complete Model • Objective Function • Maximize Z (profit) = 5x1 + 4x2 • Constraints • 6x1 + 4x224 (Raw material M1) • x1 + 2x26 (Raw material M2) • x2 2 • -x1 + x2  1 • x1, x2 0 LINEAR PROGRAMMING

LP Assumptions • Proportionality requires the contribution of each decision variable in both the objective function and the constraints to be directly proportional to the value of the variable. In the RM example, if quantity discounts are offered then the revenues will not be proportional to sales. • Additivity stipulates that the total contribution of all the variables in the objective function and in the constraints to be the direct sum of individual contribution of each variable. • Certainty stipulates that each linear coefficient of the objective function and constraints is known. • Divisibility stipulates that the variable is assumed to take fractional values. LINEAR PROGRAMMING

One More example A tape recorder company manufactures models A, B & C, which have profit contribution per unit of Rs 15, Rs 40 and Rs 60 respectively. The wekly minimum production requirements are 25 units of model A, 130 units for model B, and 55 units of model C. A dozen units of model A requires 4 hours for manufacturing, 3 hours for assembling and 1 hour for packaging. The corresponding figures for dozen units of B & C are (2.5, 4,2) and (6,9,4) respectively. During the forthcoming week the company has available 130 hours of manufacturing, 170 hours of assembling and 52 hours of packaging. Formulate the LP model. LINEAR PROGRAMMING

Solution • Ten minutes to solve. LINEAR PROGRAMMING

LINEAR PROGRAMMING