operations research

Management Science

04/18/23 Dr.Sourabh Bishnoi 1

SOURABH BISHNOIM. Sc.(Statistics). Ph. D.(Operations Research)

TITLE OF MY THESIS: Time / Cost Optimization Process Through Goal Programming Model With Special Reference To Transportation And Assignment Problem.

This research work is completed under the kind guidance of Dr. Sabir Ali Siddiqui, Head Department of Statistics, St. John’s College, Agra.04/18/23 Dr.Sourabh Bishnoi 2

Management science An approach to managerial decision making that is based on the scientific method, makes extensive use of quantitative analysis.

A variety of names exists for the body of knowledge involving quantitative approaches to decision making; in addition to management science, another widely known and accepted name is Operations Research.

Today, many use of terms operations research and management science interchangeably we shall treat them as synonyms.


The scientific management revolution of the early 1900, initiated by Frederic W. Taylor, provided the foundation for OR.


But modern OR is generally considered to have originated during World War II period, when some teams of scientists were formed to deal with strategic and tactical problems faced by the military. These teams, comprises of the specialist from diverse fields, e.g. mathematicians, engineers, behavioral scientists, etc., were joined together to solve a common problem through the utilization of scientific method. As the research was done to optimize the military operations, the name OR evolved. 04/18/23 Dr.Sourabh Bishnoi 5

After the war, many of these team members continued their research on quantitative approaches to decision making. • The continued research on quantitative

approaches to decision making resulted in numerous methodological developments.

• Probably the most significant development was the discovery by American mathematician George B. Dantzig, in 1947, of the simplex method for solving Linear Programming Problems.



Linear programming was initially referred to as “programming in a linear structure.” In 1948 Tjalling Koopmans suggested to George Dantzig that the name was much to long; Koopmans suggestion was to shorten it to Linear Programming. George Dantzig agreed and the field we now we know as linear programming was named.


Many more methodological developments followed, and in 1957 the first book* on OR was published by Churchman, Ackoff and Arnoff. *(Introduction to OR, New York: Wiley, 1957)

Linear Programming Problem:

A quantitative Approach To Decision making Of Business Situations.


MODEL DEVELOPMENTModels are representations of real objects or situations. The representations, or models, can be presented in various forms. For example a scale model of an aero plane is the representation of the real aero plane. Similarly, a child’s toy truck is a model of a real truck.


Types Of Model:


According to the structure, Models are of three types:

• Iconic Or Physical Model

• Analog Model

• Mathematical Or Symbolic Model


The model aero plane or toy truck are examples of models that are physical replicas of real object. In modeling terminology physical replicas are referred to as Iconic Models.

A second classification of models includes those that are physical in form but do not have the same physical appearance as the object being modeled. Such models are referred to as Analog Models.


The Speedometer of an automobile is an analog model; The position of the needle on the dial represents the speed of the automobile. A Thermometer is another analog model representing temperature.04/18/23 Dr.Sourabh Bishnoi 14

A third classification of models includes those that represent a problem by a system of symbols and mathematical relationships or expressions. Such models are referred to as Mathematical models Or Symbolic Model and are a critical part of any quantitative approach to decision making.


For example the total profit from the sale of a product can be determined by multiplying the profit per unit by the quantity sold. If we let x represent the number of units sold and P the total profit earned by selling 10 units:

P=10x


The purpose, or value, of any model is that it enables us to make inferences about the real situation by studying and analyzing the model.

For example, an airplane designer might test an iconic model of a new airplane in a wind tunnel to learn about the potential flying characteristics of the full size airplane.

Similarly, a mathematical model may be used to make inferences about how much profit will be earned if a specified quantity of a particular product is sold. According to the mathematical model of equation, we would expect to obtain a Rs.30 profit by selling three units of the product.


MATHEMATICAL MODELLING

Mathematical Modeling is the process of translating a verbal statement of a problem into a mathematical statement. In linear programming, the mathematical statement of a problem is a linear program. The problems that are small, are not particularly difficult to model. But, as problems become larger and more complex, some general guidelines for model formulation are useful.


LINEAR PROGRAMMING:

Linear programming is a problem solving approach that has been developed to help managers to make decisions.

Linear programming is a problem solving approach that has been developed for situations involving maximizing or minimizing a linear function subject to linear constraints that limit the degree to which the objective can be pursued. 04/18/23 Dr.Sourabh Bishnoi 19

Some typical applications of linear programming are (i)A manufacturer wants to develop a production schedule and an inventory policy that will satisfy sales demand in future periods. Ideally, the schedule and policy will enable the company to satisfy demand and at the same time minimize the total production and inventory costs.


(ii) A financial analyst must select an investment portfolio from a variety of stock and bond investment alternatives. The analyst would like to establish the portfolio that maximizes the return on investment.


(iii) A marketing manager wants to wants to determine how best to allocate a fixed advertising budget among alternative advertising media such as radio, television, newspapers, and magazines. The manager would like to determine the media mix that maximizes the advertising effectiveness.


(iv) A company has warehouses in a number of locations throughout the United States. Given a set of customer demands for its products, the company would like to determine which warehouse should ship how much product to which customers so that the total transportation costs are minimized.


These are only a few examples of situations where linear programming has been used successfully, but they illustrate the diversity of linear programming applications

A close scrutiny reveals one basic property that all of them have in common. In each example, we were concerned with maximizing or minimizing some quantity.



We wanted to minimize costs in example 1, we wanted to maximize return on investment in example 2, we wanted to maximize advertising effectiveness in example 3we wanted to minimize total transportation costs in example 4.

In all linear programming problems, the maximization or minimization of some quantity is the objective

GUIDELINES FOR MODEL

FORMULATION • The process of formulating linear

programming models is an art that can only be mastered with practice and experience. Although every problem has some unique features, most problems also have common features. As a result, some general guidelines for model formulation can be helpful, especially for beginners.


(1) Understand the problem thoroughly.

(2) Write a verbal statement of the objective function and each constraint.

(3) Define the decision variables:

(4) Write the objective function in terms of the decision variables.

(5) Write the constraints in terms of the decision variables


(1)Understand the problem thoroughly:

Read the problem description to quickly get a feel for what is involved. Identify those items that you feel should be included in the model. If the problem is especially complex, take notes; these notes will help you focus on the key ideas and facts.


(2)Write a verbal statement of the objective function and each constraint:

Later you will translate these verbal statements into mathematical statements. At this step, you might write the objective function as, for instance, maximize profit or minimize monthly operating costs. You might write a constraint limiting funds borrowed as follows: funds borrowed ≤ line of credit. Even experienced management scientists find they sometimes make mistakes when skipping this step.


(3)Define the decision variables:

Ask yourself what decisions the manager must make. What does she or he control? The decision variables should be chosen to represent these decisions. The decision variables should also be defined in such a fashion that writing a mathematical statement of the objective function and the left hand side of constraints is facilitated.


(4)Write the objective function in terms of the decision variables:

Translate your verbal statement of the objective function developed in step 2 into a mathematical statement; the mathematical statement must be a linear function of the decision variables.


(5)Write the constraints in terms of the decision variables:

Translate your verbal statement of each constraint developed in step 2 into a mathematical statement; the left hand side of the resulting equation or inequality must be a linear function of the decision variables.



THE ELECTRONIC COMMUNICATIONS PROBLEM Electronic Communications manufactures portable radio systems that can be used for two- way communications. The company’s new product, which has a range of up to 25 miles, is particularly suitable for use in a variety of business and personal applications. The distribution channels for the new radio are as follows:

Marine equipment distributorsBusiness equipment distributorsNational chain of retail storesMail order

Because of differing distribution and promotional costs, the profitability of the product will vary with the distribution channel. In addition, the advertising cost and the personal sales effort required will vary with the distribution channels. Table below summarizes the contribution to profit, advertising cost, and personal sales effort data pertaining to the

Electronic Communications problem. The firm has set the advertising budget at Rs.500000 and there is a maximum of 1800 hours of sales force time available for allocation to the sales effort. Management has also decided to produce exactly 6000 units for the current production period. Finally, an ongoing current with the national chain of retail stores requires that atleast 1500 units be distributed through this distribution channel.

Electronic Communications is now faced with the problem of establishing a strategy that will provide for the distribution of the radios in such a way that overall profitability of the new radio production will be maximized. Decisions must be made as to how many units should be allocated to each of the four distributions channels, as well as how to allocate the advertising budget and sales force effort to each of the four distribution channels.

Profit, advertising cost, and personal sales time data for the Electronic Communications problem

Distribution Channel

Profit per Unit Sold

Advertising Cost per Unit Sold

Personal Sales Effort per Unit Sold

Marine distributors

Rs. 90 Rs. 10 2 hours

Business distributors


National retail stores


Mail order Rs. 60 Rs. 15 None


FORMULATION OF THE ELECTRONIC COMMUNICATIONS PROBLEM


A written statement of the problem has been presented. Let us now attempt to write a verbal statement of the objective function and each constraint. For the objective function, we can write Objective function: Maximize profit


There appear to be four constraints necessary for the problem. They are necessary because of (1) a limiting advertising budget, (2) limited sales force availability, (3) a production requirement, and (4) a retail stores distribution requirement • Constraint 1: Advertising expenditures ≤ Budget• Constraint 2: Sales time used ≤ Time available• Constraint 3: Radios produced = Management

requirement• Constraint 4: Retail distribution ≥ Contract

requirement04/18/23 Dr.Sourabh Bishnoi 37

This provides a verbal description of the objective function and the constraints. We are now ready to define the decision variables such that they represent the decisions that the manager must make.


For the Electronic Communications problems, we introduce the following four decision variables:


• X1 = the number of units produced for the marine equipment distribution channel

• X2 = the number of units produced for the business equipment distribution channel

• X3 = the number of units produced for the national retail chain distribution channel

• X4 = the number of units produced for the mail-order distribution channel

Using the data of above table, the objective function for maximizing the total contribution to profit associated with the radios can be written as follows:

Max 90 X1 + 84 X2 + 70 X3 + 60 X4


Let us now develop a mathematical statement of the constraints for the problem. Since the advertising budget has been set at Rs. 5000, the constraint that limits the amount of advertising expenditure can be written as 10 X1 + 8 X2 + 9 X3 + 15 X4 ≤ 50000


Similarly, since the sales time is limited to 1800 hours, we obtain the constraint

2 X1 + 3 X2 + 3 X3 ≤ 1800


Management’s decision to produce exactly 6000 units during the current production period is expressed as

1 X1 + 1 X2 + 1 X3 + 1 X4 = 6000


Finally, to account for the fact that the number of units distributed by the national chain of retail stores must be Atleast 1500, we add the constraint

1 X3 ≥ 1500


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