PROJECT REPORT MCOM QUANTITATIVE TECNIQUES FOR BUSINESS

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MANUFACTURING . APPLICATION

Table Of Content

1: Introduction 02

2: Classic applications of Linear programming: 033: Uses of Linear Programming: 044: Advantages of Linear Programming: 045: Application of Linear Programming:  056: Basic requirements of a Linear Programming model: 057: Graphical method for solving a Linear Programming problem: 068: Limitations of Linear Programming model: 069: Manufacturing Application: 0810: Production Scheduling: 0911: Problem: 0912: Solution: 1013: Conclusion: 14

ABSTRACTLinear programming is the process of taking various linear inequalities relating to some situation, and finding the "best" value obtainable under those conditions. Objectives of business decisions frequently involve maximizing profit or minimizing costs. Linear programming uses linear algebraic relationships to represent a firm’s decisions, given a business objective, and resource constraints. Linear programming (LP) is a widely used mathematical technique designed to help operations managers plan and make the decisions necessary to allocate resources. In business, it is often desirable to find the production levels that will produce the maximum profit or the minimum cost. The production process can often be described with a set of linear inequalities called constraints. The profit or cost function to be maximized or minimized is called the objective function. The process of finding the optimal levels with the system of linear inequalities is called linear programming (as opposed to non-linear programming).

INTRODUCTION:Linear programming is not a programming language like C++, Java, or Visual Basic. Linear programming can be defined as:

"A method to allocate scarce resources to competing activities in an optimal manner when the problem can be expressed using a linear objective function and linear inequality constraints."

A linear program consists of a set of variables, a linear objective function indicating the contribution of each variable to the desired outcome, and a set of linear constraints describing the limits on the values of the variables. The answer" to a linear program is a set of values for the problem variables that results in the best largest or smallest value of the objective function and yet is consistent with all the constraints. Formulation is the process of translating a real-world problem into a linear program. Once a problem has been formulated as a linear program, a computer program can be used to solve the problem. In this regard, solving a linear program is relatively easy. The hardest part about applying linear programming is formulating the problem and interpreting the solution.

Decision Variables: The variables in a linear program are a set of quantities that need to be determined in order to solve the problem; i.e., the problem is solved when the best values of the variables have been identified. The variables are sometimes called decision variables because the problem is to decide what value each variable should take. Typically, the variables represent the amount of a resource to use or the level of some activity. Frequently, defining the variables of the problem is one of the hardest and/or most crucial steps in formulating a problem as a linear program. Sometimes creative variable definition can be used to dramatically reduce the size of the problem or make an otherwise non-linear problem linear.

Objective Function:The objective of a linear programming problem will be to maximize or to minimize some numerical value. This value may be the expected net present value of a project or a forest property; or it may be the cost of a project; it could also be the amount of wood produced, the expected number of visitor-days at a park, the number of endangered species that will be saved, or the

amount of a particular type of habitat to be maintained. Linear programming is an extremely general technique, and its applications are limited mainly by our imaginations and our ingenuity.

The objective function indicates how much each variable contributes to the value to be optimized in the problem.

Constraints:These are mathematical expressions that combine the variables to express limits on the possible solutions. For example, they may express the idea that the number of workers available to operate a particular machine is limited, or that only a certain amount of steel is available per day.

Classic applications of Linear programming:

1. Manufacturing:Product choice Several alternative outputs with different input requirements Scarce inputs Maximize profit.

2. Agriculture:Feed choice Several possible feed ingredients with different nutritional content Nutritional requirements Minimize costs.

3. The Transportation Problem:Several depots with various amounts of inventory Several customers to whom shipments must be made Minimize cost of serving customers. 4. Scheduling:Many possible personnel shifts Staffing requirements at various times Restrictions on shift timing and length Minimize cost of meeting staffing requirements.

5. Finance: Several types of financial instruments available Cash flow requirements over time Minimize cost.

Uses of Linear Programming:

There are many uses of L.P. It is not possible to list them all here. However L.P is very useful to find out the following: 1: Optimum product mix to maximize the profit.2: Optimum schedule of orders to minimize the total cost.3: Optimum media-mix to get maximum advertisement effect.4: Optimum schedule of supplies from warehouses to minimize transportation costs.5: Optimum line balancing to have minimum idling time.6: Optimum allocation of capital to obtain maximum R.O.I7: Optimum allocation of jobs between machines for maximum utilization of machines.8: Optimum assignments of jobs between workers to have maximum labor productivity.9: Optimum staffing in hotels, police stations and hospitals to maximize efficiency.10: Optimum number of crew in buses and trains to have minimum operating costs.11: Optimum facilities in telephone exchange to have minimum break downs.

Advantages of Linear Programming:

1: Provide the best allocation of available resources.2: Meet overall objectives of the management.3: Assist management to take proper decisions.4: Provide clarity of thought and better appreciation of problem.5: Improve objectivity of assessment of the situation.6: Put across our view points more successfully by logical argument supported by scientific methods.

Application of Linear Programming: 

The primary reason for using linear programming methodology is to ensure that limited resources are utilized to the fullest extent without any waste and that utilization is made in such a way that the outcomes are expected to be the best possible. Some of the examples of linear programming are:

a) A production manager planning to produce various products with the given resources of raw materials, man-hours, and machine-time for each product must determine how many products and quantities of each product to produce so as to maximize the total profit.

b) An investor has a limited capital to invest in a number of securities such as stocks and bonds. He can use linear programming approach to establish a portfolio of stocks and bonds so as to maximize return at a given level of risk.

c) A marketing manager has at his disposal a budget for advertisement in such media as newspapers, magazines, radio and television. The manager would like to determine the extent of media mix which would maximize the advertising effectiveness.

d) A Farm has inventories of a number of items stored in warehouses located indifferent parts of the country that are intended to serve various markets. Within the constraints of the demand for the products and location of markets, the company would like to determine which warehouse should ship which product and how much of it to each market so that the total cost of shipment is minimized.

Basic requirements of a Linear Programming model:

1: The system under consideration can be described in terms of a series of activities and outcomes. These activities (variables) must be competing with other variables for limited resources and relationships among these variables must be linear and the variables must be quantifiable.

2: The outcomes of all activities are known with certainty.3: A well defined objective function exist which can be used to evaluate different outcomes. The objective function should be expressed as a linear function of the decision variables. The purpose is to optimize the objective function which may be maximization of profits or minimization of costs, and so on.4: The resources which are to be allocated among various activities must be finite and limited.5: There must not be just a single course of action but r a number of feasible courses of action open to the decision maker, one of which would give the best result.

Graphical method for solving a Linear Programming problem:

If the linear programming problem is two variable problems, it can be solved graphically. The steps required for solving a linear programming problem by graphic method are:

1: Formulate the problem into a linear programming problem.2: Each inequality in the constraints may be written as equality.3: Draw straight lines corresponding to the equations obtained in step 2. So there will be as many straight lines as there are questions.4: Identify the feasible region. Feasible region is the area which satisfies all constraints simultaneously.5: The permissible region or feasible region is a many sided figure. The corner points of the figure are to be located and their co-ordinates are to be measured.

Limitations of Linear Programming model:

1: There is no guarantee that linear programming will give integer valued equations. For instance, solution may result in producing 8.291 cars. In such a situation, the manager will examine the possibility of producing 8 as well as 9 cars and will take a decision which ensures higher profits subject to given

constraints. Thus, rounding can give reasonably good solutions in many cases but in some situations we will get only a poor answer even by rounding. Then, integer programming techniques alone can handle such cases

2: Under linear programming approach, uncertainty is not allowed. The linear programming model operates only when values for costs, constraints etc. are known but in real life such factors may be unknown.

3: The assumption of linearity is another formidable limitation of linear programming. The objective functions and the constraint functions in the Linear programming model are all linear. We are thus dealing with a system that has constant returns to scale. In many situations, the input-output rate for an activity varies with the activity level. The constraints in real life concerning business and industrial problems are not linearly related to the variables, in most economic situations, sooner or later, the law of diminishing marginal returns begins to operate.

4: Linear programming will fail to give a solution if management has conflicting multiple goals. In Linear programming model, there is only one goal which is expressed in the objective function.

Eg. Maximizing the value of the profit function or minimizing he cost function, one should resort to Goal programming in situations involving multiple goals. All these limitations of linear programming indicate only one thing- that linear programming cannot be made use of in all business problems. Linear programming is not a panacea for all management and industrial problems. But for those problems where it can be applied, the linear programming is considered a very useful and powerful tool.

Manufacturing Application:Manufacturing problems: In these problems, we determine the number of units of different products which should be produced and sold by a firm when each product requires a fixed manpower, machine hours, labor hour per unit of product, warehouse space per unit of the output etc., in order to make maximum profit.

Production Scheduling:Setting a low-cost production schedule over a period of weeks or months is a difficult and important management problem in most plants. The production manager has to consider many factors: labor capacity, inventory and storage costs, space limitations, product demand, and labor relations. Because most companies produce more than one product, the scheduling process is often quite complex. Basically, the problem resembles the product mix model for each period in the future. The objective is either to maximize profit or to minimize the total cost (production plus inventory) of carrying out the task. Production scheduling is amenable to solution by LP because it is a problem that must be solved on a regular basis. When the objective function and constraints for a firm are established, the inputs can easily be changed each

month to provide an updated schedule.

Production Mix:A fertile field for the use of LP is in planning for the optimal mix of products to manufacture. A company must meet a myriad of constraints, ranging from financial concerns to sales demand to material contracts to union labor demands. Its primary goal is to generate the largest profit possible.

Problem:The Outdoor Furniture Corporation manufactures two products, benches and picnic tables, for use in yards and parks. The firm has two main resources: its carpenters (labor force) and a supply of redwood for use in the furniture. During the next production cycle, 1,200 hours of labor are available under a union agreement. The firm also has a stock of 3,500 feet of good quality redwood. Each bench that Outdoor Furniture produces requires 4 labor hours and 10 feet of redwood: each picnic table takes 6 labor hours and 35 feet of redwood. Completed benches will yield a profit of $9 each, and tables will result in a profit of $20 each. How many benches and tables should Outdoors Furniture produce to obtain the largest possible profit? Use graphical linear programming approach.

Solution:

Step 1:Translate the real life problem in to the linear programming model.

Labor (Hrs) Material (Redwood)

Profit

Benches 04 10 09Picnic Table 06 35 20

1200 3500

Step 2:

Introduce objective function and all the constraints along Non-negative condition upon decision variables.

Objective function:Our objective is to maximize the profit so profit function is £ = 20x + 9yConstraints:1: The labor constraint: 6x + 4y ≤ 1200

If,x = 0 , y = 300 and y = 0 , x = 200

2: The material constraint: 35x + 10y ≤ 3500If,x = 0 , y = 350 and y = 0 , x = 100

Step 3:After that we plot the given constraints into a graphical presentation.

Graph: 1

Red Line = The labor constraints

Green Line = The material constraints

Yellow Line = Optimal Point

Step 4:We investigate the feasible region.

Graph: 2

Red Line = The labor constraints

Green Line = The material constraints

Yellow Line = Optimal Point

Black Lines = Feasible Region

Step: 5with the help of feasible region, we are able to find the points on the corner that can optimize our solution.

Profit Function: £ = 20x + 9y

1: where x = 200 , y = 300Put,= 20 (0) + 9 (300 ) = 2700

2: where x = 34 , y = 278Put,= 20 (34) + 9 (278) = 3182

3: where x = 100 , y = 350Put= 20 (100) + 9 (0) = 2000

SO, know we know that the point where x= 34 and y = 278 is the point where companies profit is optimal.

Conclusion:From this project we came to a conclusion that 'Linear programming' is like a vast ocean where many methods, advantages, uses, requirements etc. can be seen. Linear programming can be done in any sectors where there is less waste and more profit. By this, the production of anything is possible through the new methods of L.P. As we had collected many data about Linear programming, we came to know more about this, their uses, advantages and requirements. Also, there are many different ways to find out the most suitable L.P. Also, we formulate an example for linear programming problem and done using the two methods simplex method and dual problem. And came to a conclusion that L.P is not just a technique but a planning the process of determining a particular plan of action from amongst several alternatives. Even there are limitations; L.P is a good technique, especially in the business sectors.