In this post you will get Linear Programming Notes for BCOM Exam CBCS Pattern. This post includes questions asked in previous exams. This chapter is very important because you will get atleast 10 marks in theory from this chapter.
Linear Programming Notes LPP [For BCOM Exam CBCS Pattern]
Introduction to Linear Programming (LPP)
In a decision-making, model formulation is important because it represents the essence of business decision problem. The term formulation is used to mean the process of converting the verbal description and numerical data into mathematical expressions which represents the relevant relationship among decision factors, objectives and restrictions on the use of resources.
Linear Programming (LP) is a particular type of technique used for economic allocation of ‘scarce’ or ‘limited’ resources, such as labour, material, machine, time, warehouse space, capital, energy, etc. to several competing activities, such as products, services, jobs, new equipment, projects, etc. on the basis of a given criterion of optimally. The phrase scarce resources mean resources that are not in unlimited in availability during the planning period. The criterion of optimality, generally is either performance, return on investment, profit, cost, utilily, time, distance, etc.
Here, the word linear refers to linear relationship among variables in a model. Thus, a given change in one variable will always cause a resulting proportional change in another variable.
For example, doubling the investment on a certain project will exactly double the rate of return. The word programming refers to modelling and solving a problem mathematically that involves the economic allocation of limited resources by choosing a particular course of action or strategy among various alternative strategies to achieve the desired objective.
George B Dantzing while working with US Air Force during World War II, developed this technique, primarily for solving military logistics problems. But now, it is being used extensively in all functional areas of management, hospitals, airlines, agriculture, military operations, oil refining, education, energy planning, pollution control, transportation planning and scheduling, research and development, etc.
Structure of Linear Programming
The general structure of LP model consists of three components.
A) Decision variables (activities): There is need to evaluate various alternatives (courses of action) for arriving at the optimal value of objective function. Obviously, if there are no alternatives to select from, there is no need of LP. The evaluation of various alternatives is guided by the nature of objective function and availability of resources. For this, one should pursue certain activities usually denoted by x1, x2…xn. The value of these activities represent the extent to which each of these is performed.
For example, in a product-mix manufacturing, the management may use LP to decide how many units of each of the product to manufacture by using its limited resources such as personnel, machinery, money, material, etc. These activities are also known as decision variables because they are under the decision-maker’s control.
These decision variables, usually interrelated in terms of consumption of limited resources, require simultaneous solutions. All decision variables are continuous, controllable and non-negative. That is, x1>0, x2>0, …. xn>0.
B) The objective function: The objective function of each L.P problem is a mathematical representation of the objective in terms of a measurable quantity such as profit, cost, revenue, distance, etc. In its general form, it is represented as:
Optimise (Maximise or Minimise) Z = c1x1 + c2x2. … cnxn
where Z is the measure-of-performance variable, which is a function of x1, x2 …, xn. Quantities c1, c2…cn are parameters that represent the contribution of a unit of the respective variable x1, x2 …, xn to the measure-of-performance Z. The optimal value of the given objective function is obtained by the graphical method or simplex method.
C) The constraints: There are always certain limitations (or constraints) on the use of resources, e.g. labour, machine, raw material, space, money, etc. that limit the degree to which objective can be achieved. Such constraints must be expressed as linear equalities or inequalities in terms of decision variables. The solution of an L.P model must satisfy these constraints.
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Requirements for LP Model
The linear programming 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 linear mathematical functions. In order to apply linear programming, there are certain requirements to be met:
a) There should be an objective which should be clearly identifiable and measurable in quantitative terms. It could be, for example, maximisation of sales, of profit, minimisation of cost, and so on.
b) The activities to be included should be distinctly identifiable and measurable in quantitative terms, for instance, the products included in a production planning problem.
c) The resources of the system which arc to be allocated for the attainment of the goal should also be identifiable and measurable quantitatively. They must be in limited supply. The technique would involve allocation of these resources in a manner that would trade off the returns on the investment of the resources for the attainment of the
d) The relationships representing the objective as also the resource limitation considerations, represented by the objective function and the constraint equations or inequalities, respectively must be linear in nature.
e) There should be a series of feasible alternative courses of action available to the decision makers, which are determined by the resource constraints.
Assumptions of Linear Programming
The following five basic assumptions are necessary for all linear programming models.
a) Certainty: In all LP models, it is assumed, that all model parameters such as availability of resources, profit (or cost) contribution of a unit of decision variable and consumption of resources by a unit of decision variable must be known and is constant. The linear programming is obviously deterministic in nature.
b) Divisibility (or continuity): The solution values of decision variables and resources are assumed to have either whole numbers (integers) or mixed numbers (integer and fractional). However, if only integer variables are desired, e.g. machines, employees, etc. the integer programming method may be applied to get the desired values.
c) Additivity: The value of the objective function for the given values of decision variables and the total sum of resources used, must be equal to the sum of the contributions (profit or cost) earned from each decision variable and the sum of the resources used by each decision variable, respectively. This assumption implies that
there is no interaction among the decision variables.
d) Finite choices: A linear programming model also assumes that a limited number of choices are available to a decision-maker and the decision variables do not assume negative values. Thus, only non-negative levels of activity are considered feasible. This assumption is indeed a realistic one.
e) Linearity (or proportionality): All relationships in the LP model (i.e. in both objective function and constraints) must be linear. For example, if production of one unit of a product uses 5 hours of a particular resource, then making 3 units of that product uses 3 x 5 = 15 hours of that resource.
Advantages of Linear Programming
Following are certain advantages of linear programming.
a) Linear programming helps in attaining the optimum use of productive resources. It also indicates how a decision-maker can employ his productive factors effectively by selecting and distributing (allocating) these resources.
b) Linear programming techniques improve the quality of decisions. The decision-making approach of the user of this technique becomes more objective and less subjective.
c) Linear programming techniques provide possible and practical solutions since there might be other constraints operating outside the problem which must be taken into account.
d) Highlighting of bottlenecks in the production processes is the most significant advantage of this technique. For example, when a bottleneck occurs, some machines cannot meet demand while other remains idle for some of the time.
e) Linear programming also helps in re-evaluation of a basic plan for changing conditions. If conditions change when the plan is partly carried out, they can be determined so as to adjust the remainder of the plan for best results.
Limitations of Linear Programming
In spite of having many advantages and wide areas of applications, there are some limitations associated with this technique. These are given below:
a) Linear programming treats all relationships among decision variables as linear. However, generally, neither the objective functions nor the constraints in real-life situations concerning business and industrial problems are linearly related to the variables.
b) While solving an LP model, there is no guarantee that we will get integer valued solutions. For example, in finding out how many men and machines would be required to perform a particular job, a non-integer valued solution will be meaningless. Rounding off the solution to the nearest integer will not yield an optimal solution. In such cases, integer programming is used to ensure integer value to the decision variables.
c) Linear programming model does not take into consideration the effect of time and uncertainty. Thus, the LP model should be defined in such a way that any change due to internal as well as external factors can be incorporated.
d) Sometimes large-scale problems can be solved with linear programming techniques even when assistance of computer is available. For it, the main problem can be fragmented into several small problems and solving each one separately.
e) Parameters appearing in the model are assumed to be constant but in real-life situations, they are frequently neither known nor constant.
f) It deals with only single objective, whereas in real-life situations we may come across conflicting multi-objective problems. In such cases, instead of the LP model, a goal programming model is used to get satisfactory values of these objectives.
Applications areas of Linear Programming
Linear programming is the most widely used technique of decision-making in business and Industry and in various other fields. In this section, we will discuss a few of the broad application areas of linear programming.
è Agricultural Applications
These applications fall into categories of farm economics and farm management. The former deals with agricultural economy of a nation or region, while the latter is concerned with the problems of the individual farm. Linear programming can be applied in agricultural planning, e.g. allocation of limited resources such as acreage, labour, water supply and working capital, etc. in a way so as to maximise net revenue.
è Military Applications
Military applications include the problem of selecting an air weapon system against enemy so as to keep them pinned down and at the same time minimising the amount of aviation gasoline used. A variation of the transportation problem that maximises the total tonnage of bombs dropped on a set of targets and the problem of community defence against disaster, the solution of which yields the number of defence units that should be used in a given attack in order to provide the required level of protection at the lowest possible cost.
è Production Management
(i) Product mix: A company can produce several different products, each of which requires the use of limited production resources. In such cases, it is essential to determine the quantity of each product to be produced knowing its marginal contribution.
(ii) Production planning: This deals with the determination of minimum cost production plan over planning period of an item with a fluctuating demand, considering the initial number of units in inventory, production capacity, constraints on production, manpower and all relevant cost factors. The objective is to minimise total operation costs.
(iii) Assembly-line balancing: This problem is likely to arise when an item can be made by assembling different components. The process of assembling requires some specified sequence(s). The objective is to minimise the total elapse time.
(iv) Blending problems These problems arise when a product can be made from a variety of available raw materials, each of which has a particular composition and price. The objective here is to determine the minimum cost blend, subject to availability of the raw materials, and minimum and maximum constraints on certain product constituents.
è Financial Management
(i) Portfolio selection This deals with the selection of specific investment activity among several other activities.
(ii) Profit planning This deals with the maximisation of the profit margin from investment in plant facilities and equipment, cash in hand and inventory.
è Personnel Management
a) Staffing problem Linear programming is used to allocate optimum manpower to a particular job so as to minimise the total overtime cost or total manpower.
b) Determination of equitable salaries Linear programming technique has been used in determining equitable salaries and sales incentives.
c) Job evaluation and selection: Selection of suitable person for a specified job and evaluation of job in organisations has been done with the help of linear programming technique.
Other applications of linear programming lie in the area of administration, education, fleet utilisation, awarding contracts, hospital administration and capital budgeting, etc.
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