LPP-Introduction & Formulation

LPP

• A large number of decision problems faced by a business manager involves allocation of resources to various activities, with the objective of increasing profits or decreasing costs, or both.

• When resources are in excess, no difficulty is experienced.

• Practically the mangers are confronted with the problem of scarce resources.

• The decision problem becomes complicated when a number of resources are required and there are several activities to perform.

LPP Contd.

• Mathematical programming involves optimization of a certain function, called the objective function subject to certain constraints.

• For example, a manager may be faced with the problem of deciding the appropriate product mix of the four products. With the profitability of the products along with their requirements row materials, labor, etc known, his problem can be formulated as mathematical programming problem taking the objective as the max. of profits obtainable from the mix, keeping in view the various constraints-availability of raw materials, labor supply, market and so on.

Some Applications

• A manufacturer wants to develop a production schedule and an inventory policy that will satisfy sales demand in future periods.

• A financial analyst must select an investment portfolio from a variety of stocks and bond investment alternatives.

• A marketing manager wants to determine how best to allocate a fixed advertising budget among alternative advertising media such as radio, TV, newspaper etc.

Some Applications

• A company has 4 warehouses in a number of locations throughout the state. For a set of customer demands, company would like to determine how much each warehouse should ship to each customer.

• In all LPPs the maximization or minimization of some quantity is the objective.

• A second property is called restrictions or constraints that limit the degree to which the objective can be pursued. Thus constraints are another general feature of every LPP.

LPP Contd.

• The LP method is a technique for choosing the best

alternative from a set of feasible alternatives, in

situations in which the objective function as well as

the constraints can be expressed as a linear

mathematical function.

• It includes the objective function, the constraints

and non-negativity condition.

General LPP

• Max. Z = c1x1+c2x2+……+cnxn (Objective Function) Subject to

a11x1+ a12x2+…..+ a1nxn ≤ b1

a21x1+ a22x2+…..+ a2nxn ≤ b2 (Constraints) etc

am1x1+ am2x2+…..+ amnxn ≤ bm

x1, x2, ……., xn≥ 0 (Non-negativity condition)

where cj,aij and bi ( i = 1, 2, ….., m ; j = 1, 2, ….., n) are

known constants and xj’s are decision variables and respectively termed as the profit coefficients, technological coefficients, and resource values.

General LPP

• In matrix notation, an LPP can be written as

Max Z = cX Min Z = cX Subject to AX ≤ B Subject to AX ≥ B X ≥ 0 X ≥ 0Where c = row matrix containing the coefficients in the objective function. X = column matrix containing decision variables. A = m x n matrix containing the coefficients in the constraints. B = column matrix containing the RHS values of the constraints.

The Par, Inc Problem

Par, Inc., is a small manufacturer of golf equipment and supplies whose management has decided to move into the market for medium-and high-priced golf bags. Par’s distributor is enthusiastic about the new product line and has agreed to buy all

bags Par produces over the next 3 months. After a thorough investigation of the steps involved in the manufacturing a golf bag, management has determined that each golf bag produced will require the following operations.

1. Cutting and Dyeing 2. Sewing 3. Finishing and

4. Inspection and Packaging

Par Inc.. • After analyzing each operations the director of

manufacturing concluded that if the company produces a medium-priced standard model, each bag will require 7/10 hr, 1/2hr,1hr,and 1/10hr respectively in the above mentioned departments. Similarly the high-priced deluxe model will require 1hr,5/6hr,2/3hr,and 1/4hr respectively n the above departments. After studying the departmental work

load projections, the director estimates that 630hrs,600hrs,708hrs, and 135hrs will be available for the production of golf bags during the next 3 months respectively in the above departments.

Par Inc… The accounting department has analyzed the

production data ,and arrived at prices for both bags that will result in a profit contribution of $10 for every standard bag and $9 for every deluxe bag produced.

• Question: Develop a mathematical model that can be used to determine the number of standard bags and the number of deluxe bags to produce in order to maximize total profit contribution?

Mathematical model for Par Inc.

• Let S = Number of standard bags

D = Number of deluxe bags

S,D are called decision variables.

Max. 10S + 9D

Subject to (s.t) the constraints

0.7S + 1D ≤ 630 (Cutting & Dyeing)

0.2S + 0.83D ≤ 600 (Sewing)

1S + 0.33D ≤ 708 (Finishing)

0.1S + 0.25D ≤ 135 (Inspection & Packaging)

S,D ≥ 0 (Non-negativity)

Note

• The above mathematical model is called the LP model for the Par, Inc., The special feature that makes a mathematical model a linear program is that the objective function and all the constraint functions are linear functions of decision variables.

• The word ‘programming’ means ‘choosing a course of action’

• Linear programming involves a course of action when mathematical model of the problem contains only linear functions.

LPP(Conti…)

• Feasible solution - a vector x which satisfies the constraints of an LP. The set of feasible solutions is called the feasible set or feasible region. The value of the objective function at a feasible solution x is denoted by VOF(x).

• Optimal Solution. A feasible solution that makes the value of the objective function an optimum (maximum or minimum) is called an optimal solution (maximizer, minimizer).

Problem-1

• A firm is engaged in producing two products A and B. each unit of product A requires 2kg of raw materials and 4 labor hrs for processing, whereas each unit of product B requires 3kg material and 3 hrs of labor of the same type. Every week, the firm has an availability of 60kg raw materials and 96 labor hrs. One unit of product A sold yields Rs 40 and one unit of product B sold gives Rs 35 as profit. Formulate this problem as a LPP to determine as to how many units of each of the products should be produced per week so that the firm can earn the max. profit. Assume that there is no marketing constraint so that all that is produced can be sold.