**ABSTRACT**

In the area of linear programming problem, modeling of transportation problem is fundamental in solving most real life problems as far as optimization is concerned.

There are several methods and analytic techniques for solving transportation problems.

In this project three methods (North West corner method, Minimum cost method and Vogel’s approximation method) have been used to find the initial basic feasible solution for the balanced transportation model. The results of the three methods were noted to be different.

It is noted that the Vogel’s approximation has the least transportation cost.

**CHAPTER ONE**

**1.0 INTRODUCTION**

**1.1 OPERATION RESEARCH (HISTORICAL BACKROUND)**

The word operation research is referred to as operational research by the British/Europeans and the Americans referred to it as operation research, but both are often shortened or abbreviated to ‘OR’ which is the most commonly used term, another term which is used for this term is ‘management science’ (MS). The Americans sometimes combine terms OR and MS together and say “OR/MS” or “ORMS” and yet other terms sometimes used are “INDUSTRIAL ENGINEERING” (IE) and decision science (DC) in recent years, there have been a move towards the standardization upon a single term for the field namely the term “ OR”. Operation research tools are not centralized on any specific discipline but it takes its tools from different discipline such as mathematics, Statistics, Economics, and Psychology, Engineering e.t.c and then combines these tools to create a new set of knowledge for decision making. Recently O.R became a professional discipline which deals with the application of science methods for making decision and especially to the allocation of scarce resources.

The main purpose of operation research is to a rational basis for decision making in the absence of complete information simply because the system is composed of human, machine and procedures may not have complete information, O.R specialist are involved in three classical aspects of science which are stated as follows

- Determine the system behavior
- Analyzing the systems behavior by developing appropriate models.
- Predicts future behavior based on the models.

The emphasis on analysis of operations as a whole distinguishes O.R from other research and engineering based on the fact that O.R is an interdisciplinary discipline which provide solutions to problems of military operations during world war II and as well successful in other operations but today business applications are primarily concerned with O.R analysis for the possible alternative solutions.

Operation research is a field of applied sciences or a scientific methodology (Mathematical, analytical, and quantitative) which by assessing the overall implication of various alternative courses of action in a management system provides an impress basis for management decisions. In a nut-shell, O.R is a field of sciences that deals with the applications of advanced analytical methods to help in making a vital decision and often considered to be a subfield of mathematics employing techniques from mathematical analysis, mathematical modeling, statistical analysis and management optimization theory.

**1.2 BRIEF HISTORY OF OPERATION RESEARCH **

Operation research is a relatively new discipline which began in a systematic fashion in the late 1930’s and started in the UK, early in 1936, the British air ministry established Bawdsey research station on the east coast near Felixstowe, Suffolk as the centre where all prewar radar experiments for both the airforce and the Army would be carried out. Experimental radar equipment was brought up to a high state of reliability and ranges of over 100 miles on aircraft were obtained. Royal Airforce (RAF) fighter commander, in the same year 1936 charged specifically with then air defense of Britain was first created and however lacked any effective aircraft. No hurricanes or spit fives had come into service and no radar was yet fed into its elementary warning and control system.

In the summer of 1937, the first of three major pre-war air defense exercises was carried out. The experimental radar station at Bawdsey research station was brought into operation and the information derived from it was fed into the general air-defense warning and control system. From the early warning point of view, this exercise was encouraging but the tracking information obtained from radar after filtering and transmission through the control and display network was not very satisfactory.

In July 1938, a second major air-defense exercise was carried out and four additional radar stations had been installed along the coast and it was hoped that Britain now had an aircraft location and controlled system greatly improved both in coverage and effectiveness. Accordingly, on the termination of the exercise, the superintendent of Bawdsey research station, A.P Rowe, announced that although the exercise had again demonstrated the technical feasibility of the radar system for detecting aircraft, its operational achievement still fell far short of requirements. He therefore, proposed that a crash program of research into the operation-as opposed to the technical aspects of the system should begin immediately. The term “operational Research” (Research into (military) operations) was coined as a suitable description of this new branch of applied sciences, and the first team was selected amongst the scientists of the radar research group same day.

In the summer of 1939, Britain held what is assumed to be its last pre-war air-defense exercise. It involves some 33,000 men, 1300 aircrafts, 110 aircraft guns, 700 search light and 100 barrage balloons. This exercise projected a great improvement in the operation of the air defense warning and control system. The contributions made by the OR team was so apparent that the air officer commander in-chief RAF fighter commander (Air chief Marshall, sin Hugh Dowding) requested that on the outbreak of war, they should be attached to his headquarter at Stanmore in north London. On May 15^{th} 1940, with German forces advancing rapidly in France, Stanmore research section was requested to analyze a French request for ten additional fighter squadrons (12 aircraft a squadron, meaning that we have 120 aircraft in all) when losses were running at some three squadrons every two days (i.e. 36 aircraft every two days). They prepared graphs for Wintson Churchill (the British prime minister of the time) based upon a study of current daily losses and replacements rates, indicating how rapidly such a move will deplete fighters strength. No aircraft were sent and most of those currently in France we recalled.

In 1941, an operational Research section (ORS) was established in coastal command which was to carry out some of the most well-known OR moved in World War II. The responsibility of coastal command was to a large extent, the flying of long-range sorties by single aircraft with the object of sighting and attacking surfaced U-boats (unlike modern day submarines)surfacing was necessary to recharge batteries, vent the boat of fumes and recharge air tanks. Movement of the u-boats were much faster on the surface than underwater as well as being less easily detected by sonar, thus, the operation research started just before World war II in Britain with the establishment of teams of scientists to study the strategic and tactical problems involved in military operations. The objectives is to find the most effective utilization of limited military resources by the use of quantitative techniques. Following the end of the war, OR spread even though it spreads in different ways in the UK and USA.

In 1951, a committee on operation research formed by the national research council of USA and the first book on “Methods of operations research” by Morse and Kinball was published and in 1952 the operations research society of America came into existence. Success of operation research in army attracted the attention of industrial managers who were seeking solutions to their complex business problems. Now a days, almost every organization in all countries have staff applying operation research and the use of operation research in government has spread from military to wide variety of departments at all levels. The growth of operation research is not limited to the USA and UK, it has reached many countries of the world.

India was one of the few first countries that started using operations research, in India, Regional research laboratory created at Hydewabad was the first operation research unit established during 1949. In 1953, operation research unit was established in Indian statistical institute Calcutta with the objective of using operations research method in national planning and survey. In 1955, operations research society of India was formed, which happened to be one of the first members of International Federation of Operations Research Societies. Today, operations research is a popular subject in management institutes and schools of mathematics.

**1.3 TERMINOLOGIES USED IN OPERATIONS RESEARCH**

**MODEL: -** This is a mathematical representation of real life problems that reproduces the essential elements of the system being studied. (British Standard Institution, 1995).

**PARAMETERS: - **This is a value describing a problem that is constant within a process but may be changed for subsequent process. (British Standard Institution, 1995).

**SYSTEM: - **This is a coherent and logical arrangement of principles, data, material, components, equipments and or procedures (B.S.I 1995).

**DATA: - **These are facts and statistics collected together for reference or analysis (British Standard Institution, 1995).

**CONSTRAINTS: - **This is an equation or inequality relating the variables in an optimization problem (British Standard Institution, 1995).

**FEASIBLE SOLUTION: - **A solution satisfying the constraints of a mathematical programming program (British Standard Institution, 1995).

**INFEASIBLE SOLUTION: - **A solution to a mathematical programming problem where constraints, specifically the non-negative constraints of the problem have not all been satisfied (British Standard Institution, 1995).

**BASIC FEASIBLE SOLUTION: - **A basic solution that satisfies the constraints of a linear programming problem including the non-negative constraints (British Standard Institution, 1995).

**BASIC SOLUTION: - **A solution to a linear programming problem where the number of variable with non-zero values is at most equal to the number of constraints in the problem. (British Standard Institution, 1995).

**BASIC: - **An men non-negative matrix consisting of in-column of the problem matrix (British Standard Institution, 1995).

**OBJECTIVE FUNCTION: - **The combination of the variables of a mathematical programming problem whose value is to be maximized subject to the constraints of the problem (British Standard Institution, 1995).

**OPTIMAL SOLUTION: - **A set of values of variables in a mathematical programming problem that optimize the assigned constraints (British Standard Institution, 1995).

**SLACK VARIABLE: - **A linear constraints of the form can be converted into an equality by adding a new non-negative variable to the left hand side of the inequality. Such variable is numerically equal to the left and right hand side of the inequality and it is known as slack variable. It represents the waste involved in the phase of the system modeled (S.O Adewale, 2018).

**SURPLUS VARIABLE: - **A linear constraints of the form can be converted to an equality by subtracting a new non-negative variable from the left hand side of the inequality, such variable is known as surplus variable. It simply represents excess input into the phase of the modeled system by the constraints. (S.O Adewale, 2018).

**ARTIFICIAL VARIABLE: - **A new variable added to the left hand side of each constraints that contains a surplus variable (S.O Adewale, 2018).

**ITERATION: - **A single cycle of operation in the algorithm used in solving a problem. (British Standard Institution, 1995).

**DECISION VARIABLE: - **A quantity that varies, which is under control of the decision maker. (British Standard Institution, 1995).

**OPTIMIZATION: - **This is the act of obtaining the best possible solution within the constraints of a problem. (British Standard Institution, 1995).

**1.4 TOOLS AND TECHNIQUES USED IN O.R **

Operation research uses any suitable tools or techniques available. The most common and frequently used tools or techniques are mathematical procedures, cost analysis, electronic computation. However, operations researchers give special importance to the development and use of techniques like linear programming, game theory, decision theory, queuing theory, inventory models and simulation. In addition to the above techniques, some other common tools are non-linear programming, inter programming, dynamic programming, sequencing theory, marker process, network scheduling, symbolic model, information theory and value theory . The brief explanations of some of the tools or techniques are as follows;

**LINEAR PROGRAMMING: -**This is a constrained optimization technique which optimizes some criterion within some constraints. In linear programming the objective function (profits, loses or return on investment) and constraints and linear. (Adewale, 2018)**NON-LINEAR PROGRAMMING: -**This is used when the objective function and the constraints are not linear in nature. Linear relationship may be applied to approximate non-linear constraints but limited to some range because approximation becomes poorer as the range is extended. Thus, the non-linear programming is used to determine the approximation in which a solution lies and then the solution is obtained using linear methods.**DYNAMIC PROGRAMMING: -**Dynamic programming is a method of analyzing multistage decision processes. In this, each elementary decision depends on those proceeding decisions and as well as external factors.**INTEGER PROGRAMMING: -**This is the phenomenon where one or more variables of the problem takes integral values on the dynamic programming are used. For example, number of Motors in an organization, number of passengers in an aircraft, number of generators in a power generating plants. e.t.c**MARKOV PROCESS: -**Markov process permits to predict changes over time, information about the behavior of a system is known. This is used in decision making in situations where the various states are defined. The probability from one state to another state is known and depends on the current state and is independent of how we have arrived at that particular state.**INFORMATION THEORY: -**This analytical process is transferred from the electrical communication field to O.R field. The objective of this theory is to evaluate the effectiveness of flow of information with a given system. This is used mainly in communication networks but also has indirect influence in simulating the examination of business organizational structure with a review of enhancing flow of information.

**1.5 APPLICATIONS OF OPERATION RESEARCH **

Today almost all fields of business and government are utilizing the benefits of operations research. There are voluminous applications of operations research. Although, it is not feasible to cover all applications of O.R. in brief. The following are abbreviated sets of typical operations research applications to show how widely these techniques are used today:

**ACCOUNTING: -**Assigning audit teams effectively, credit policy analysis, cash flow planning, standard costs development, planning of delinquent account strategy.**CONSTRUCTION: -**Project scheduling, monitoring and controls determination of proper workforce, deployment of workforce, allocation of resources to projects.**FACILITIES PLANNING: -**Factory location and size decision, estimation of number of facilities required, hospital planning, international logistic system design, transportation loading and unloading, ware house location decision.**FINANCE: -**Building cash management models, allocating capital among various alternatives, building finance planning models, investment models, portfolio analysis, dividend policy making.**MANUFACTURING: -**Inventory control, marketing balance projection, production scheduling, and production smoothing.**MARKETING: -**Advertising budget allocation, product introduction timing, selection of product number, decision most effective packaging alternative.

**1.6 LINEAR PROGRAMMING PROBLEM FORMULATION**

The linear programming formulation is illustrated through a product mix problem. The product mix problem occurs in an industry where it is possible to manufacture a variety of products. A product has a certain margin of profit per unit and uses a common pool of limited resources. In this case, the linear programming techniques identify the product combination which will minimize the profit subject to the availability of limited resources constraints.

**METHODS OF SOLVING LINEAR PROGRAMMING PROBLEMS**

The following methods are used in solving linear programming problems.

- Graphical Method.
- Simplex Method.