Q.1. What is a linear programming problem ? Discuss the steps and role of linear programming is solving management problems. Discuss and describe the role of liner programming in managerial decision-making bringing out limitations, if any.
Ans : Linear Programming is a mathematical technique useful for allocation of scarce or limited resources to several competing activities on the basis of given criterion of optimality.
The usefulness of linear programming as a tool for optimal decision-making on resource allocation, is based on its applicability to many diversified decision problems. The effective use and application requires, as on its applicability to many diversified decision problems. The effective use and application requires, as a first step, the mathematical formulation of an LP model, when the problem is presented in words. Steps of linear programming model formulation are summarized as follows :
STEP 1 : Identify the Decision Variables
a) Express each constraint in words. For this you should first see whether the constraint is of the form >/ (at least as large as), of the form \< (no larger than) or of the form = (exactly equal to)
b) You should then verbally express the objective function
c) Steps (a) and (b) should then allow you to verbally identify the decision variables
If there are several decision alternatives available , then in order to identify the decision variables you need to ask yourself the question – what decisions must be made in order to optimize the objective function ?
Having accomplished step 1(a) through (c) decide the symbolic notation for the decision variables and specify units of measurement. Such specification of units of measurement would help in interpreting the final solution of the LP problem .
STEP 2 : Identify the Problem Data
For solving a problem, we need to identify the problem data so as to provide the actual values for the decision variables. For this, we need to enumerate all types of information with respect to the given problem in order to determine the values of these decision variables. These quantities constitute the problem data. It may be noted that the decision-maker can control values of the variables but cannot control the values of the data
STEP 3 : Formulate the constraints
a) Verbally express the constraints in terms of the requirements and availability of each resource.
b) Convert the verbal expression of the constraints imposed by the resource availability as linear equality or inequality , in terms of the decision variables defined in step 1
These constraints are the conditions in that the decision variable must satisfy in order to constitute an acceptable (feasible) solution. These constraints typically arise due to physical limitations, management imposed restrictions , external restrictions , logical restrictions on individual variables ,implied relationships among variables etc. Wrong formulation can either lead to solutions that are not feasible ir ti tge exclusion of some solutions that are actually feasible and possibly optimal.
STEP 4 : Formulate the objective function
Identify the objective function is to be maximized or minimized. Then express it verbally - maximize total profit/cost and then conver it into a linear mathematical expression in terms of decision variables multiplied by their profit or cost contributions
After having enough experiences in model building one may skip verbal description, the following are certain examples of LP model formulation that you can use to strengthen your ability to translate a real – life problem into a mathematical model .
Linear Programming used in Marketing management
• Media Selection : The linear programming technique helps in determining the advertising media mix so as to maximize the effective exposure, subject to limitation of budget, specified exposure rates to different market segments....
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