Unit 1

Introduction to Operational Research

Lesson 1: Introduction to Operational Research Introduction to Operational Research This teaching module is designed to be an entertaining and representative introduction to the subject of Operational Research. It is divided into a number of sections each covering different aspects of OR.

What is Operational Research?

This looks at the characteristics of Operational Research, how you define what OR is and why organisations might use it. It considers the scientific nature of OR and how it helps in dealing with problems involving uncertainty, complexity and conflict.

OR is the representation of real-world systems by mathematical models together with the use of quantitative methods (algorithms) for solving such models, with a view to optimising.

Battle of the Atlantic

Considers the origins of OR in the British military and looks at how OR helped to ensure the safety of merchant ships during the "Battle of the Atlantic" in World War II.

Introduction to OR

Terminology

The British/Europeans refer to "operational research", the Americans to "operations research" - but both are often shortened to just "OR" (which is the term we will use).

Another term which is used for this field is "management science" ("MS"). The Americans sometimes combine the terms OR and MS together and say "OR/MS" or "ORMS". Yet other terms sometimes used are "industrial engineering" ("IE") and "decision science" ("DS"). In recent years there has been a move towards a standardisation upon a single term for the field, namely the term "OR".

Books

There are many books on OR available in the college library and you should not need to buy any books. If you do find you need a book then I recommend:

J.K.Sharma: Operations Research (Theory and Application)

N.D.Vohra: Quantitative Techniques in Management

Journals

OR is a new field which started in the late 1930's and has grown and expanded tremendously in the last 30 years (and is still expanding). As such the academic journals contain many useful articles that reflect state of the art applications of OR. We give below a selection of the major OR journals.

1. Operations Research 2. Management Science 3. European Journal of Operational Research 4. Journal of the Operational Research Society 5. Mathematical Programming 6. Networks 7. Naval Research Logistics 8. Interfaces

The first seven of the above are mainly theoretical whilst the eighth (Interfaces) concentrates upon case studies. All of these journals are available in the college library so have a browse through them to see what is happening in state of the art OR.

Note here that my personal view is that in OR, as in many fields, the USA is the country that leads the world both in the practical application of OR and in advancing the theory (for example, the American OR conferences have approximately 2500 participants, the UK OR conference has 300).

One thing I would like to emphasise in relation to OR is that it is (in my view) a subject/discipline that has much to offer in making a real difference in the real world. OR can help you to make better decisions and it is clear that there are many, many people and companies out there in the real world that need to make better decisions. I have tried to include throughout OR-Notes discussion of some of the real-world problems that I have personally been involved with.

History of OR

OR is a relatively new discipline. Whereas 70 years ago it would have been possible to study mathematics, physics or engineering (for example) at university it would not have been possible to study OR, indeed the term OR did not exist then. It was really only in the late 1930's that operational research began in a systematic fashion, and it started in the UK. As such I thought it would be interesting to give a short history of OR and to consider some of the problems faced (and overcome) by early OR workers.

Whilst researching for this short history I discovered that history is not clear cut, different people have different views of the same event. In addition many of the participants in the events described below are now elderly/dead. As such what is given below is only my understanding of what actually happened.

Note: some of you may have moral qualms about discussing what are, at root, more effective ways to kill people. However I cannot change history and what is presented

below is essentially what happened, whether one likes it or not.

1936

Early in 1936 the British Air Ministry established Bawdsey Research Station, on the east coast, near Felixstowe, Suffolk, as the centre where all pre-war radar experiments for both the Air Force 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.

It was also in 1936 that Royal Air Force (RAF) Fighter Command, charged specifically with the air defense of Britain, was first created. It lacked however any effective fighter aircraft - no Hurricanes or Spitfires had come into service - and no radar data was yet fed into its very elementary warning and control system.

It had become clear that radar would create a whole new series of problems in fighter direction and control so in late 1936 some experiments started at Biggin Hill in Kent into the effective use of such data. This early work, attempting to integrate radar data with ground based observer data for fighter interception, was the start of OR.

1937

The first of three major pre-war air-defence exercises was carried out in the summer of 1937. 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.

1938

In July 1938 a second major air-defense exercise was carried out. Four additional radar stations had been installed along the coast and it was hoped that Britain now had an aircraft location and control system greatly improved both in coverage and effectiveness. Not so! The exercise revealed, rather, that a new and serious problem had arisen. This was the need to coordinate and correlate the additional, and often conflicting, information received from the additional radar stations. With the out-break of war apparently imminent, it was obvious that something new - drastic if necessary - had to be attempted. Some new approach was needed.

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 achievements still fell far short of requirements. He therefore proposed that a crash program of research into the operational - 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 science. The first team was selected from amongst the scientists of the radar research group the same day.

1939

In the summer of 1939 Britain held what was to be its last pre-war air defence exercise. It involved some 33,000 men, 1,300 aircraft, 110 antiaircraft guns, 700 searchlights, and 100 barrage balloons. This exercise showed a great improvement in the operation of the air defence warning and control system. The contribution made by the OR teams was so apparent that the Air Officer Commander-in-Chief RAF Fighter Command (Air Chief Marshal Sir Hugh Dowding) requested that, on the outbreak of war, they should be attached to his headquarters at Stanmore in north London.

Initially, they were designated the "Stanmore Research Section". In 1941 they were redesignated the "Operational Research Section" when the term was formalised and officially accepted, and similar sections set up at other RAF commands.

1940

On May 15th 1940, with German forces advancing rapidly in France, Stanmore Research Section was asked to analyse a French request for ten additional fighter squadrons (12 aircraft a squadron - so 120 aircraft in all) when losses were running at some three squadrons every two days (i.e. 36 aircraft every 2 days). They prepared graphs for Winston Churchill (the British Prime Minister of the time), based upon a study of current daily losses and replacement rates, indicating how rapidly such a move would deplete fighter strength. No aircraft were sent and most of those currently in France were recalled.

1941 onward

In 1941 an Operational Research Section (ORS) was established in Coastal Command which was to carry out some of the most well-known OR work 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 (German submarines). Amongst the problems that ORS considered were:

• organisation of flying maintenance and inspection

Here the problem was that in a squadron each aircraft, in a cycle of approximately 350 flying hours, required in terms of routine maintenance 7 minor inspections (lasting 2 to 5 days each) and a major inspection (lasting 14 days). How then was flying and maintenance to be organised to make best use of squadron resources?

ORS decided that the current procedure, whereby an aircrew had their own aircraft, and that aircraft was serviced by a devoted ground crew, was inefficient (as it meant that when the aircraft was out of action the aircrew were also inactive). They proposed a central garage system whereby aircraft were sent for maintenance when required and each aircrew drew a (different) aircraft when required.

The advantage of this system is plainly that flying hours should be increased. The disadvantage of this system is that there is a loss in morale as the ties between the aircrew and "their" plane/ground crew and the ground crew and "their" aircrew/plane are broken.

This is held by some to be the most strategic contribution to the course of the war made by OR (as the aircraft and pilots saved were consequently available for the successful air defense of Britain, the Battle of Britain).

The first use of OR techniques in India, was in the year] 1949 at Hyderabad,

where at the Regional Research Institute, an independent operations research unit was

set-up. To identify evaluate and solve the problems related to planning, purchases and

proper maintenance of stores, an operations research unit was also setup at the Defence

Science Laboratory use of OR tools and techniques was done during India's second five

year Plan in demand forecasting and suggesting the most suitable scheme which would

lead to the overall growth and the development of the economy. Even today, Planning

Commission utilises some. of these techniques in framing policies and sector-wise

performance evaluation.

In 1953, at the Indian Statistical Institute (Calcutta). A self-sufficient operations

research unit was established for the purpose of national planning & survey. OR Society

of India was folined in 1957 which publishes journal titled "Of search". Many big and

prominent business & industrial houses are using extensively the tools of OR for the

optimum utilisation of precious and scarce resources available to them. This phenomenon

is not limited to the private sector only. Even good companies in the public sector ("Nav

Ratnas") are reaping the benefits of fully functional sound OR units. Example of such

corporate, both private & public are: SAIL, BHEL, NTPC, Indian Railways, Indian

Airlines, Air-India, Hindustan Lever, TELCO & TISCO etc. Textile companies engaged

in the process of manufacture of various types of fabrics use some of the tools of OR like

lenear programming & PERT in their blending, dyeing and other manufacturing

operations.

Various other Indian companies are .employing OR techniques for solving problems

pertaining to various spheres of activities, as diverse as advertising, sales promotion,

inspection, quality control, staffing, personnel, investment & production planning, etc.

These organisations are not only employing the operations research techniques

and analysis on a short-term trouble-shooting basis but also for ong-range strategic

planning.

Basic OR concepts

Definition

So far we have avoided the problem of defining exactly what OR is. In order to get a clearer idea of what OR is we shall actually do some by considering the specific problem below and then highlight some general lessons and concepts from this specific example.

Two Mines Company

The Two Mines Company own two different mines that produce an ore which, after being crushed, is graded into three classes: high, medium and low-grade. The company has contracted to provide a smelting plant with 12 tons of high-grade, 8 tons of medium-grade and 24 tons of low-grade ore per week. The two mines have different operating characteristics as detailed below.

Mine Cost per day (£'000) Production (tons/day) High Medium Low X 180 6 3 4 Y 160 1 1 6

How many days per week should each mine be operated to fulfil the smelting plant contract?

Note: this is clearly a very simple (even simplistic) example but, as with many things, we have to start at a simple level in order to progress to a more complicated level.

Guessing

To explore the Two Mines problem further we might simply guess (i.e. use our judgement) how many days per week to work and see how they turn out.

• work one day a week on X, one day a week on Y

This does not seem like a good guess as it results in only 7 tonnes a day of high-grade, insufficient to meet the contract requirement for 12 tonnes of high-grade a day. We say that such a solution is infeasible.

• work 4 days a week on X, 3 days a week on Y

This seems like a better guess as it results in sufficient ore to meet the contract. We say that such a solution is feasible. However it is quite expensive (costly).

Rather than continue guessing we can approach the problem in a structured logical fashion as below. Reflect for a moment though that really we would like a solution which supplies what is necessary under the contract at minimum cost. Logically such a minimum cost solution to this decision problem must exist. However even if we keep guessing we can never be sure whether we have found this minimum cost solution or not. Fortunately our structured approach will enable us to find the minimum cost solution.

Two Mines solution

What we have is a verbal description of the Two Mines problem. What we need to do is to translate that verbal description into an equivalent mathematical description.

In dealing with problems of this kind we often do best to consider them in the order:

1. variables 2. constraints 3. objective.

We do this below and note here that this process is often called formulating the problem (or more strictly formulating a mathematical representation of the problem).

(1) Variables

These represent the "decisions that have to be made" or the "unknowns".

Let

x = number of days per week mine X is operated y = number of days per week mine Y is operated

Note here that x >= 0 and y >= 0.

(2) Constraints

It is best to first put each constraint into words and then express it in a mathematical form.

• ore production constraints - balance the amount produced with the quantity required under the smelting plant contract

Ore High 6x + 1y >= 12 Medium 3x + 1y >= 8 Low 4x + 6y >= 24

Note we have an inequality here rather than an equality. This implies that we may produce more of some grade of ore than we need. In fact we have the general rule: given a choice between an equality and an inequality choose the inequality.

For example - if we choose an equality for the ore production constraints we have the three equations 6x+y=12, 3x+y=8 and 4x+6y=24 and there are no values of x and y which satisfy all three equations (the problem is therefore said to be "over-constrained"). For example the values of x and y which satisfy 6x+y=12 and 3x+y=8 are x=4/3 and y=4, but these values do not satisfy 4x+6y=24.

The reason for this general rule is that choosing an inequality rather than an equality gives us more flexibility in optimising (maximising or minimising) the objective (deciding values for the decision variables that optimise the objective).

• days per week constraint - we cannot work more than a certain maximum number of days a week e.g. for a 5 day week we have

x

Constraints of this type are often called implicit constraints because they are implicit in the definition of the variables.

(3) Objective

Again in words our objective is (presumably) to minimise cost which is given by 180x + 160y

Hence we have the complete mathematical representation of the problem as:

minimise 180x + 160y subject to 6x + y >= 12 3x + y >= 8 4x + 6y >= 24 x

usually be ignored (note that often the data (numbers) that we use in formulating the LP will be inaccurate anyway).

• the way the complete mathematical representation of the problem is set out above is the standard way (with the objective first, then the constraints and finally the reminder that all variables are >=0).

Discussion

Considering the Two Mines example given above:

• this problem was a decision problem

• we have taken a real-world situation and constructed an equivalent mathematical representation - such a representation is often called a mathematical model of the real-world situation (and the process by which the model is obtained is called formulating the model). Just to confuse things the mathematical model of the problem is sometimes called the formulation of the problem.

• having obtained our mathematical model we (hopefully) have some quantitative method which will enable us to numerically solve the model (i.e. obtain a numerical solution) - such a quantitative method is often called an algorithm for solving the model.

Essentially an algorithm (for a particular model) is a set of instructions which, when followed in a step-by-step fashion, will produce a numerical solution to that model. You will see some examples of algorithms later in this course..

• our model has an objective, that is something which we are trying to optimise.

• having obtained the numerical solution of our model we have to translate that solution back into the real-world situation.

Hence we have a definition of OR as:

OR is the representation of real-world systems by mathematical models together with the use of quantitative methods (algorithms) for solving such models, with a view to optimising.

One thing I wish to emphasise about OR is that it typically deals with decision problems. You will see examples of the many different types of decision problem that can be tackled using OR.

We can also define a mathematical model as consisting of:

• Decision variables, which are the unknowns to be determined by the solution to the model.

• Constraints to represent the physical limitations of the system. • An objective function. • A solution (or optimal solution) to the model is the identification of a set of

variable values which are feasible (i.e. satisfy all the constraints) and which lead to the optimal value of the objective function.

Philosophy

In general terms we can regard OR as being the application of scientific methods/thinking to decision making. Underlying OR is the philosophy that:

• decisions have to be made; and • using a quantitative (explicit, articulated) approach will lead (on average) to better

decisions than using non-quantitative (implicit, unarticulated) approaches (such as those used (?) by human decision makers).

Indeed it can be argued that although OR is imperfect it offers the best available approach to making a particular decision in many instances (which is not to say that using OR will produce the right decision).

Often the human approach to decision making can be characterised (conceptually) as the "ask Fred" approach, simply give Fred ('the expert') the problem and relevant data, shut him in a room for a while and wait for an answer to appear.

The difficulties with this approach are:

• speed (cost) involved in arriving at a solution • quality of solution - does Fred produce a good quality solution in any particular

case • consistency of solution - does Fred always produce solutions of the same quality

(this is especially important when comparing different options).

You can form your own judgement as to whether OR is better than this approach or not.

Phases of an OR project

Drawing on our experience with the Two Mines problem we can identify the phases that a (real-world) OR project might go through.

1. Problem identification

• Diagnosis of the problem from its symptoms if not obvious (i.e. what is the problem?)

• Delineation of the subproblem to be studied. Often we have to ignore parts of the entire problem.

• Establishment of objectives, limitations and requirements.

2. Formulation as a mathematical model

It may be that a problem can be modelled in differing ways, and the choice of the appropriate model may be crucial to the success of the OR project. In addition to algorithmic considerations for solving the model (i.e. can we solve our model numerically?) we must also consider the availability and accuracy of the real-world data that is required as input to the model.

Note that the "data barrier" ("we don't have the data!!!") can appear here, particularly if people are trying to block the project. Often data can be collected/estimated, particularly if the potential benefits from the project are large enough.

You will also find, if you do much OR in the real-world, that some environments are naturally data-poor, that is the data is of poor quality or nonexistent and some environments are naturally data-rich. As examples of this church location study (a data-poor environment) and an airport terminal check- in desk allocation study (a data-rich environment).

This issue of the data environment can affect the model that you build. If you believe that certain data can never (realistically) be obtained there is perhaps little point in building a model that uses such data.

3. Model validation (or algorithm validation)

Model validation involves running the algorithm for the model on the computer in order to ensure:

• the input data is free from errors • the computer program is bug-free (or at least there are no outstanding bugs) • the computer program correctly represents the model we are attempting to

validate • the results from the algorithm seem reasonable (or if they are surprising we can at

least understand why they are surprising). Sometimes we feed the algorithm historical input data (if it is available and is relevant) and compare the output with the historical result.

4. Solution of the model

Standard computer packages, or specially developed algorithms, can be used to solve the model (as mentioned above). In practice, a "solution" often involves very many solutions

under varying assumptions to establish sensitivity. For example, what if we vary the input data (which will be inaccurate anyway), then how will this effect the values of the decision variables? Questions of this type are commonly known as "what if" questions nowadays.

Note here that the factors which allow such questions to be asked and answered are:

• the speed of processing (turn-around time) available by using pc's; and • the interactive/user-friendly nature of many pc software packages.

5. Implementation

This phase may involve the implementation of the results of the study or the implementation of the algorithm for solving the model as an operational tool (usually in a computer package).

In the first instance detailed instructions on what has to be done (including time schedules) to implement the results must be issued. In the second instance operating manuals and training schemes will have to be produced for the effective use of the algorithm as an operational tool.

It is believed that many of the OR projects which successfully pass through the first four phases given above fail at the implementation stage (i.e. the work that has been done does not have a lasting effect). As a result one topic that has received attention in terms of bringing an OR project to a successful conclusion (in terms of implementation) is the issue of client involvement. By this is meant keeping the client (the sponsor/originator of the project) informed and consulted during the course of the project so that they come to identify with the project and want it to succeed. Achieving this is really a matter of experience.

A graphical description of this process is given below.

The phases that a typical OR project might go through are:

1. problem identification 2. formulation as a mathematical model 3. model validation 4. solution of the model 5. implementation

We would be looking for a discussion of these points with reference to one particular problem.

Example OR projects

Not all OR projects get reported in the literature (especially OR projects which fail). However to give you an idea of the areas in which OR can be applied we give below some abstracts from papers on OR projects that have been reported in the literature (all projects drawn from the journal Interfaces).

Note here that, at this stage of the course, you will probably not understand every aspect of these abstracts but you should have a better understanding of them by the end of the course.

• Yield management at American Airlines

Critical to an airline's operation is the effective use of its reservations inventory. American Airlines began research in the early 1960's in managing revenue from this inventory. Because of the problem's size and difficulty, American Airlines Decision Technologies has developed a series of OR models that effectively reduce the large problem to three much smaller and far more manageable subproblems: overbooking, discount allocation and traffic management. The results of the subproblem solutions are combined to determine the final inventory levels. American Airlines estimates the quantifiable benefit at $1.4 billion over the last three years and expects an annual revenue contribution of over $500 million to continue into the future. Yield management is also sometimes referred to as capacity management. It applies in systems where the cost of operating is essentially fixed and the focus is primarily, though not exclusively, on revenue maximisation. For example all transport systems (air, land, sea) operating to a fixed timetable (schedule) could potentially benefit from yield management. Hotels would be another example of a system where the focus should primarily be on revenue maximisation.

To give you an illustration of the kind of problems involved in yield management suppose that we consider a specific flight, say the 4pm on a Thursday from Chicago O'Hare to New York JFK. Further suppose that there are exactly 100 passenger seats on the plane subdivided into 70 economy seats and 30 business class seats (and that this subdivision cannot be changed). An economy fare is $200 and a business class fare is $1000. Then a fundamental question (a decision problem) is :

How many tickets can we sell ?

One key point to note about this decision problem is that it is a routine one, airlines need to make similar decisions day after day about many flights.

Suppose now that at 7am on the day of the flight the situation is that we have sold 10 business class tickets and 69 economy tickets. A potential passenger phones up requesting an economy ticket. Then a fundamental question (a decision problem) is : Would you sell it to them ? Reflect - do the figures given for fares $200 economy, $1000 business affect the answer to this question or not ?

Again this decision problem is a routine one, airlines need to make similar decisions day after day, minute after minute, about many flights. Also note that in this decision problem an answer must be reached quickly. The potential passenger on the phone expects an immediate answer. One factor that may influence your thinking here is consider certain money (money we are sure to get) and uncertain money (money we may, or may not, get).

Suppose now that at 1pm on the day of the flight the situation is that we have sold 30 business class tickets and 69 economy tickets. A potential passenger phones up

requesting an economy ticket. Then a fundamental question (a decision problem) is : Would you sell it to them ?

• NETCAP - an interactive optimisation system for GTE telephone network planning

With operations extending from the east coast to Hawaii, GTE is the largest local telephone company in the United States. Even before its 1991 merger with Contel, GTE maintained more than 2,600 central offices serving over 15.7 million customer lines. It does extensive planning to ensure that its $300 million annual investment in customer access facilities is well spent. To help GTE Corporation in a very complex task of planning the customer access network, GTE Laboratories developed a decision support tool called NETCAP that is used by nearly 200 GTE network planners, improving productivity by more than 500% and saving an estimated $30 million per year in network construction costs.

• Managing consumer credit delinquency in the US economy: a multi-billion dollar management science application

GE Capital provides credit card services for a consumer credit business exceeding $12 billion in total outstanding dollars. Its objective is to optimally manage delinquency by improving the allocation of limited collection resources to maximise net collections over multiple billing periods. We developed a probabilistic account flow model and statistically designed programs to provide accurate data on collection resource performance. A linear programming formulation produces optimal resource allocations that have been implemented across the business. The PAYMENT system has permanently changed the way GE Capital manages delinquent consumer credit, reduced annual losses by approximately $37 million, and improved customer goodwill.

Note here that GE Capital also operates in the UK.

Operational research example 1987 UG exam

Managing director of the company started as a tea-boy 40 years ago and rose through the ranks of the company (without any formal education) to his present position. He believes that all a person needs to succeed in business are (innate) ability and experience. What arguments would you use to convince him that the decision-making techniques dealt with in this course are of value?

Solution

The points that we would expect to see in an answer include:

• OR obviously of value in tactical situations where data well defined • an advantage of explicit decision making is that it is possible to examine

assumptions explicitly

• might expect an "analytical" approach to be better (on average) than a person • OR techniques combine the ability and experience of many people • sensitivity analysis can be performed in a systematic fashion • OR enables problems too large for a person to tackle effectively to be dealt with • constructing an OR model structures thought about what is/is not important in a

problem • a training in OR teaches a person to think about problems in a logical fashion • using standard OR techniques prevents a person having to "reinvent the wheel"

each time they meet a suitable problem • OR techniques enable computers to be used with (usually) standard packages and

consequently all the benefits of computerised analysis (speed, rapid (elapsed) solution time, graphical output, etc)

• OR techniques an aid (complement) to ability and experience not a substitute for them

• many OR techniques simple to understand and apply • there have been many successful OR projects (e.g. ...) • other companies use OR techniques - do we want to be left behind? • ability and experience are vital but need OR to use these effectively in tackling

large problems • OR techniques free executive time for more creative tasks

Unit 1

Linear Programming

Lesson 2: Introduction to linear programming And

Problem formulation

Definition And Characteristics Of Linear Programming

Linear Programming is that branch of mathematical programming which is

designed to solve optimization problems where all the constraints as will as the objectives

are expressed as Linear function .It was developed by George B. Denting in 1947. Its

earlier application was solely related to the activities of the second’ World War. However

soon its importance was recognized and it came to occupy a prominent place in the

industry and trade.

Linear Programming is a technique for making decisions under certainty i.e.;

when all the courses of options available to an organisation are known & the objective of

the firm along with its constraints are quantified. That course of action is chosen out of

all possible alternatives which yields the optimal results. Linear Programming can also be

used as a verification and checking mechanism to ascertain the accuracy and the

reliability of the decisions which are taken solely on the basis of manager's experience-

without the aid of a mathematical model.

Some of the definitions of Linear Programming are as follows:

"Linear Programming is a method of planning and operation involved in

the construction of a model of a real-life situation having the following elements:

(a) Variables which denote the available choices and

(b) the related mathematical expressions which relate the variables to the

controlling conditions, reflect clearly the criteria to be employed for measuring the

benefits flowing out of each course of action and providing an accurate measurement of

the organization’s objective. The method maybe so devised' as to ensure the selection of

the best alternative out of a large number of alternative available to the organization

Linear Programming is the analysis of problems in which a Linear function of a

number of variables is to be optimized (maximized or minimized) when whose variables

are subject to a number of constraints in the mathematical near inequalities.

From the above definitions, it is clear that:

(i) Linear Programming is an optimization technique, where the underlying

objective is either to maximize the profits or to minim is the Cosp.

(ii) It deals with the problem of allocation of finite limited resources amongst

different competiting activities in the most optimal manner.

(iil) It generates solutions based on the feature and characteristics of the actual

problem or situation. Hence the scope of linear programming is very wide as it

finds application in such diverse fields as marketing, production, finance &

personnel etc.

(iv) Linear Programming has be-en highly successful in solving the following types

of problems :

(a) Product-mix problems

(b) Investment planning problems

(c) Blending strategy formulations and

(d) Marketing & Distribution management.

(v) Even though Linear Programming has wide & diverse’ applications, yet all LP

problems have the following properties in common:

(a)The objective is always the same (i.e.; profit maximization or cost

minimization).

(b) Presence of constraints which limit the extent to which the objective can be

pursued/achieved.

(c) Availability of alternatives i.e.; different courses of action to choose from, and

(d) The objectives and constraints can be expressed in the form of linear relation.

(VI) Regardless of the size or complexity, all LP problems take the same form i.e.

allocating scarce resources among various compete ting alternatives. Irrespective

of the manner in which one defines Linear Programming, a problem must have

certain basic characteristics before this technique can be utilized to find the

optimal values.

The characteristics or the basic assumptions of linear programming are as follows:

1. Decision or Activity Variables & Their Inter-Relationship. The decision or

activity variables refer to any activity which are in competition with other variables for

limited resources. Examples of such activity variables are: services, projects, products

etc. These variables are most often inter-related in terms of utilization of the scarce

resources and need simultaneous solutions. It is important to ensure that the relationship

between these variables be linear.

2. Finite Objective Functions. A Linear Programming problem requires a clearly

defined, unambiguous objective function which is to be optimized. It should be capable

of being expressed as a liner function of the decision variables. The single-objective

optimization is one of the most important prerequisites of linear programming. Examples

of such objectives can be: cost-minimization, sales, profits or revenue maximization &

the idle-time minimization etc

3. Limited Factors/Constraints. These are the different kinds of limitations on the

available resources e.g. important resources like availability of machines, number of man

hours available, production capacity and number of available markets or consumers for

finished goods are often limited even for a big organisation. Hence, it is rightly said that

each and every organisation function within overall constraints 0 11 both internal and

external.

These limiting factors must be capable of being expressed as linear equations or in equations in terms of decision variables 4. Presence of Different Alternatives. Different courses of action or alternatives should be available to a decision maker, who is required to make the decision which is the most effective or the optimal.

For example, many grades of raw material may be available, the’ same raw material can be purchased from different supplier, the finished goods can be sold to various markets, production can be done with the help of different machines. 5. Non-Negative Restrictions. Since the negative values of (any) physical quantity has no meaning, therefore all the variables must assume non-negative values. If some of the variables is unrestricted in sign, the non-negativity restriction can be enforced by the help of certain mathematical tools – without altering the original informati9n contained in the problem. 6. Linearity Criterion. The relationship among the various decision variables must be directly proportional Le.; Both the objective and the constraint,$ must be expressed in terms of linear equations or inequalities. For example. if--6ne of the factor inputs (resources like material, labour, plant capacity etc.) Creases, then it should result in a proportionate manner in the final output. These linear equations and in equations can graphically be presented as a straight line. 7. Additively. It is assumed that the total profitability and the total amount of each resource utilized would be exactly equal to the sum of the respective individual amounts. Thus the function or the activities must be additive - and the interaction among the activities of the resources does not exist. 8. Mutually Exclusive Criterion. All decision parameters and the variables are assumed to be mutually exclusive In other words, the occurrence of anyone variable rules out the simultaneous occurrence of other such variables. 9. Divisibility. Variables may be assigned fractional values. i.e.; they need not necessarily always be in whole numbers. If a fraction of a product can not be produced, an integer programming problem exists.

Thus, the continuous values of the decision variables and resources must be permissible in obtaining an optimal solution. 10. Certainty. It' is assumed that conditions of certainty exist i.e.; all the relevant parameters or coefficients in the Linear Programming model are ful1y and completely known and that they don't change during the period. However, such an assumption may not hold good at all times. 11. Finiteness. 'Linear Programming assumes the presence of a finite number of activities and constraints without which it is not possible to obtain the best or the optimal solution. Advantages & Limitations Of Linear Programming

Advantages of Linear Programming .Following are some of the advantages of Linear Programming approach:

1. Scientific Approach to Problem Solving. Linear Programming is the application of scientific approach to problem solving. Hence it results in a better and true picture of the problems-which can then be minutely analysed and solutions ascertained.

2. Evaluation of All Possible Alternatives. Most of the problems faced by the present organisations are highly complicated - which can not be solved by the traditional approach to decision making. The technique of Linear Programming ensures that’ll possible solutions are generated - out of which the optimal solution can be selected.

3. Helps in Re-Evaluation. Linear Programming can also be used in .re-evaluation of a basic plan for changing conditions. Should the conditions change while the plan is carried out only partially, these conditions can be accurately determined with the help of Linear Programming so as to adjust the remainder of the plan for best results.

4. Quality of Decision. Linear Programming provides practical and better quality of decisions’ that reflect very precisely the limitations of the system i.e.; the various restrictions under which the system must operate for the solution to be optimal. If it becomes necessary to deviate from the optimal path, Linear Programming can quite easily evaluate the associated costs or penalty.

5. Focus on Grey-Areas. Highlighting of grey areas or bottlenecks in the production process is the most significant merit of Linear Programming. During the periods of bottlenecks, imbalances occur in the production department. Some of the machines remain idle for long periods of time, while the other machines are unable toffee the demand even at the peak performance level.

6. Flexibility. Linear Programming is an adaptive & flexible mathematical technique and hence can be utilized in analyzing a variety of multi-dimensional problems quite successfully.

7. Creation of Information Base. By evaluating the various possible alternatives in the light of the prevailing constraints, Linear Programming models provide an important database from which the allocation of precious resources can be don rationally and judiciously.

8. Maximum optimal Utilization of Factors of Production. Linear Programming helps in optimal utilization of various existing factors of production such as installed capacity,. labour and raw materials etc.

Limitations of Linear Programming. Although Linear Programming is a highly successful having wide applications in business and trade for solving optimization' problems, yet it has certain demerits or defects.

Some of the important-limitations in the application of Linear Programming are as follows: 1. Linear Relationship. Linear Programming models can be successfully applied only in those situations where a given problem can clearly be represented in the form of linear relationship between different decision variables. Hence it is based on the implicit assumption that the objective as well as all the constraints or the limiting factors can be stated in term of linear expressions - which may not always hold good in real life situations. In practical business problems, many objective function & constraints can not

be expressed linearly. Most of \he business problems can be expressed quite easily in the form of a quadratic equation (having a power 2) rather than in the terms of linear equation. Linear Programming fails to operate and provide optimal solutions in all such cases.

e.g. A problem capable of being expressed in the form of: ax2+bx+c = 0 where a # 0 can not be solved with the help of Linear

Programming techniques.

2. Constant Value of objective & Constraint Equations. Before a Linear Programming technique could be applied to a given situation, the values or the coefficients of the objective function as well as the constraint equations must be completely known. Further, Linear Programming assumes these values to be constant over a period of time. In other words, if the values were to change during the period of study, the technique of LP would loose its effectiveness and may fail to provide optimal solutions to the problem.

However, in real life practical situations often it is not possible to determine the coefficients of objective function and the constraints equations with absolute certainty. These variables in fact may, lie on a probability distribution curve and hence at best, only the Iikelil1ood of their occurrence can be predicted. Mover over, often the value’s change due to extremely as well as internal factors during the period of study. Due to this, the actual applicability of Linear Programming tools may be restricted. 3. No Scope for Fractional Value Solutions. There is absolutely no certainty that the solution to a LP problem can always be quantified as an integer quite often, Linear Programming may give fractional-varied answers, which are rounded off to the next integer. Hence, the solution would not be the optimal one. For example, in finding out 'the pamper of men and machines required to perform a particular job, a fractional Larson-integer solution would be meaningless. 4. Degree Complexity. Many large-scale real life practical problems can not be solved by employing Linear Programming techniques even with the help of a computer due to highly complex and Lengthy calculations. Assumptions and approximations are required to be made so that $e, given problem can be broken down into several smaller problems and, then solve separately. Hence, the validity of the final result, in all such cases, may be doubtful: 5. Multiplicity of Goals. The long-term objectives of an organisation are not confined to a single goal. An organisation ,at any point of time in its operations has a multiplicity of goals or the goals hierarchy - all of which must be attained on a priority wise basis for its long term growth. Some of the common goals can be Profit maximization or cost minimization, retaining market share, maintaining leadership position and providing quality service to the consumers. In cases where the management has conflicting, multiple goals, Linear Programming model fails to provide an optimal solution. The reason being that under Linear Programming techniques, there is only one goal which can be expressed in the objective function. Hence in such circumstances, the situation or the given problem has to be solved by the help of a different mathematical programming technique called the "Goal Programming".

6. Flexibility. Once a problem has been properly quantified in terms of objective function and the constraint equations and the tools of Linear Programming are applied to it, it becomes very difficult to incorporate any changes in the system arising on account of any change in the decision parameter. Hence, it lacks the desired operational flexibility. Mathematical model of LPP. Linear Programming is a mathematical technique for generating & selecting the optimal or the best solution for a given objective function. Technically, Linear Programming may be formally defined as a method of optimizing (i.e.; maximizing or minimizing) a linear function for a number of constraints stated in the form of linear in equations. Mathematically the problem of Linear Programming may be stated as that of the optimization of linear objective function of the following form : nnii XCxCxcxCZ ++++= .....................2211 Subject to the Linear constrains of the form: 1ln1.313212111 ...................... bxaxaxaxaxa nii

≤≥+++++

222.32322211 ...................... bxaxaxaxaxa nniij≤≥+++++

1.33322211 ...................... bmxaxamxaxaxa njniij≤≥+++++

≤

mnmnii bXaxamxamnamnam ≤+++ ...................332211 These are called the non-negative constraints. From the above, it is linear that a LP problem has: (I) linear objective function which is to be maximized or minimized. (ii) various linear constraints which are simply the algebraic statement of the

limits of the resources or inputs at the disposal. (iii) Non-negatively constraints.

Linear Programming is one of the few mathematical tools that can be used to provide solution to a wide variety of large, complex managerial problems.

For example, an oil refinery can vary its product-mix by its choice among the different grades of crude oil available from various parts of the world. Also important is the process selected since parameters such as temperature would also affect the yield. As prices and demands vary, a Linear Programming model recommends which inputs and processes to use in order to maximize the profits.

Livestock gain in value as they grow, but the rate of gain depends partially on the feed choice of the proper combination of ingredients to maximize the net gain. This value can be expressed in terms of Linear Programming. A firm which distributes products over a large territory faces an unimaginable number of different choices in deciding how best to meet demand from its network of godown and warehouses. Each warehouse has a very limited number of items and demands often can not be met from the nearest warehouse. If their are 25 warehouses and 1,000 customers, there are 25,000 possible match ups

between customer and warehouse. LP can quickly recommend the shipping quantities and destinations so as to minimize the cost of total distribution.

These are just a few of the managerial problems that have been addressed successfully by LP. A few others are described throughout this text. Project scheduling can be improved by allocating funds appropriately among the most critical task so as to most effectively reduce the overall project duration. Production planning over a year or more can reduce costs by careful timing of the use of over time and inventory to control changes in the size of the workforce. In the short run, personnel work schedules must take into consideration not only the production, work preferences for day offs and absenteeism etc.

Besides recommending solutions to problems like these, LP can provide useful information for managerial decisions, that can be solved by Linear Programming. The application, however, rests on certain postulates and assumptions which have to hold good for the optimality of the solution to be effective during the planning period.

Applications Of Linear Programming Techniques In Indian Economy

In a third world developing country like India, the various factors of productions such as skilled labour, capital and raw material etc. are very precious and scarce. The policy planner is, therefore faced with the problem of scarce resource allocation to meet the various competing demands and numerous conflicting objectives. The traditional and conventional methods can no longer be applied in the changed circumstances for solving this problem and are hence fast losing their importance in the current economy. Hence, the planners in our country are continuously and constantly in search of highly objective and result oriented techniques for sound and proper decision making which can be effective at all levels of economic planning. Nonprogrammed decisions consist of capacity expansion, plant location, product line diversification, expansion, renovation and modernization etc. On the other hand, the programmed decisions consist of budgeting, replacement, procurement, transportation and maintenance etc.

In These modern times, a number of new and better methods ,techniques and tools have been developed by the economists all over the globe. All these findings form the basis of operations research. Some of these well-known operations research techniques have been successfully applied in Indian situations, such as: business forecasting, inventory models - deterministic and probabilistic, Linear Programming.Goal programming, integer programming and dynamic programming etc.

The main applications of the Linear Programming techniques, in Indian context are as follows:

1. Plan Formulation. In the formulation of the country's five year plans, the Linear Programming approach and econometric models are being used in various diverse areas such as : food grain storage planning, transportation, multi-level planning at the national, state and district levels and urban systems.

2. Railways. Indian Railways, the largest employer in public sector undertakings, has successfully applied the methodology of Linear Programming in various key areas.

For example, the location of Rajendra Bridge over the Ganges linking South Bihar and North Bihar in Mokama in preference to other sites has been achieved only by the help of Linear Programming.

3. Agriculture Sector. Linear Programming approach is being extensively used in agriculture also. It has been tried on a limited scale for the crop rotation mix of cash crops, food crops and to/ascertain the optimal fertilizer mix.

4. Aviation Industry. Our national airlines are also using Linear Programming in the selection of routes and allocation of air-crafts to various chosen routes. This has been made possible by the application of computer system located at the headquarters. Linear Programming has proved to be a very useful tool in solving such problems. '

5. Commercial Institutions. The commercial institutions as well as the individual traders are also using Linear Programming techniques for cost reduction and profit maximization. The oil refineries are using this technique for making effective and optimal blending or mixing decisions and for the improvement of finished products.

6. Process Industries. Various process industries such as paint industry makes decisions pertaining to the selection of the product mix and locations of warehouse for distribution etc. with the help of Linear Programming techniques. This mathematical technique is being extensively used by highly reputed corporations such as TELCO for deciding what castings and forging to be manufactured in own plants and what should be purchased from outside suppliers. '

7. Steel Industry. The major steel plants are using Linear Programming techniques for determining the optimal combination of the final products such as : billets, rounds, bars, plates and sheets.

8. Corporate Houses. Big corporate houses such as Hindustan Lever employ these techniques for the distribution of consumer goods throughout the country. Linear Programming approach is also used for capital budgeting decisions such as the selection of one project from a number of different projects. Main Application Areas Of Linear Programming In the last few decades since 1960s, no other mathematical tool or technique has had such a profound impact on the management's decision making criterion as Linear Programming well and truly it is one of the most important decision making tools of the last century which has transformed the way in which decisions are made and businesses are conducted. Starting with the Second World War till the Y -2K problem in computer applications, it has covered a great distance.

We discuss below some of the important application areas of Linear Programming:

I. Military Applications. Paradoxically the most appropriate example of an organization is the military and worldwide, Second World War is considered to be one of the best managed or organized events in the history of the mankind. Linear Programming is extensively used in military operations. Such applications include the problem of selecting an air weapon system against the enemy so as to keep them pinned down and at the same time minimizes the amount of fuel used. Other examples are dropping of bombs

on pre-identified targets from aircraft and military assaults against localized terrorist outfits.

2. Agriculture. Agriculture applications fall into two broad categories, farm economics and farm management. The former deals with the agricultural economy of a nation or a region, while the latter is concerned with the problems of the individual form. Linear Programming can be gainfully utilized for agricultural planning e:g. allocating scarce limited resources such as capital, factors of production like labour, raw material etc. in such a way 'so as to maximize the net revenue.

3. Environmental Protection. Linear programming is used to evaluate the various possible a1temative for handling wastes and hazardous materials so as to satisfy the stringent provisions laid down by the countries for environmental protection. This technique also finds applications in the analysis of alternative sources of energy, paper recycling and air cleaner designs.

4. Facilities Location. Facilities location refers to the location nonpublic health care facilities (hospitals, primary health centers) and’ public recreational facilities (parks, community hal1s) and other important facilities pertaining to infrastructure such as telecommunication booths etc. The analysis of facilities location can easily be done with the help of Linear Programming. Apart from these applications, LP can also be used to plan for public expenditure and drug control. '

5. Product-Mix. The product-mix of a company is the existence of various products that the company can produce and sell. However, each product in the mix requires finite amount of limited resources. Hence it is vital to determine accurately the quantity of each product to be produced knowing their profit margins and the inputs required for producing them. The primary objective is to maximize the profits of the firm subject to the limiting factors within which it has to operate.

6. Production. A manufacturing company is quite often faced with the situation where it can manufacture several products (in different quantities) with the use of several different machines. The problem in such a situation is to decide which course of action will maximize output and minimize the costs. Another application area of Linear Programming in production is the assembly by-line balancing - where a component or an item can be manufactured by assembling different parts. In such situations, the objective of a Linear Programming model is to set the assembly process in the optimal (best possible) sequence so that the total elapsed time could be minimized.

7. Mixing or Blending. Such problems arise when the same product can be produced with the help of a different variety of available raw-materials each having a fixed composition and cost. Here the objective is to determine the minimum cost blend or mix (Le.; the cost minimizations) and the various constraints that operate are the availability of raw materials and restrictions on some of the product constituents.

8. Transportation & Trans-Shipment. Linear Programming models are employed to determine the optimal distribution system i.e.; the best possible channels of distribution available to an organisation for its finished product sat minimum total cost of transportation or shipping from company's godown to the respective markets. Sometimes the products are not transported as finished products but are required to be manufactured at various. sources. In such a

situation, Linear Programming helps in ascertaining the minimum cost of producing or manufacturing at the source and shipping it from there.

9. Portfolio Selection. Selection of desired and specific investments out of a large number of investment' options available10 the managers (in the form of financial institutions such as banks, non-financial institutions such as mutual funds, insurance companies and investment services etc.) is a very difficult task, since it requires careful evaluation of all the existing options before arriving at C\ decision. The objective of Linear Programming, in such cases, is to find out the allocation/which maximizes the total expected return or minimizes the total risk under different situations.

10. Profit Planning & Contract. Linear Programming is also quite useful in profit planning and control. The objective is to maximize the profit margin from investment in the plant facilities and machinery, cash on hand and stocking-hand.

11. Traveling Salesmen Problem. Traveling salesmen problem refers to the problem of a salesman to find the shortest route originating from a particular city, visiting each of the specified cities and then returning back to the originating point of departure. The restriction being that no city must be visited more than once during a particular tour. Such types of problems can quite easily be solved with the help of Linear Programming.

12. Media Selection/Evaluation. Media selection means the selection of the optimal media-mix so as to maximise the effective exposure. The various constraints in this case are: Budget limitation, different rates for different media (i.e.; print media, electronic media like radio and T.V. etc.) and the minimum number of repeated advertisements (runs) in the various media. The use of Linear Programming facilities like the decision making process.

13. Staffing. Staffing or the man-power costs are substantial for a typical organisation which make its products or services very costly. Linear Programming techniques help in allocating the optimum employees (man-power or man-hours) to the job at hand. The overall objective is to minimize the total man-power or overtime costs.

14. Job Analysis. Linear Programming is frequently used for evaluation of jobs in an organisation and also for matching the right job with the right worker.

15. Wages and Salary Administration. Determination of equitable salaries and various incentives and perks becomes easier with the application of Linear Programming. LP tools” can also be utilized to provide optimal solutions in other areas of personnel management such as training and development and recruitment etc.

Linear Programming problem Formation Steps In Formulating A Linear Programming Model Linear programming is one of the most useful techniques for effective decision making. It is an optimization approach with an emphasis on providing the optimal solution for resource allocation. How best to allocate the scarce organisational or national resources among different competing and conflicting needs (or uses) forms the core of its working. The scope for application of linear programming is very wide and it occupies a central place in many diversified decisional problems. The effective use and application of linear programming requires the formulation of a realistic model which represents accurately

the objectives of the decision making subject to the constraints in which it is required to be made.

The basic steps in formulating a linear programming model are as follows: Step I. Identification of the decision variables. The decision variables (parameters) having a bearing on the decision at hand shall first be identified, and then expressed or determined in the form of linear algebraic functions or in equations.

Step II. Identification of the constraints. All the constraints in the given problem which restrict the operation of a firm at a given point of time must be identified in this stage. Further these constraints should be broken down as linear functions in terms of the pre-defined decision variables.

Step III. Identification of the objective. In the last stage, the objective which is required to be optimized (i.e., maximized or memorized) must be dearly identified and expressed in terms of the pre-defined decision variables.

Example 1

High Quality furniture Ltd. Manufactures two products, tables & chairs. Both the products have to be processed through two machines Ml & M2 the total machine-hours available are: 200 hours ofM1 and 400 hours of M2 respectively. Time in hours required for producing a chair and a table on both the machines is as follows:

Time in Hours

Machine Table Chair

MJ 7 4 M2 5 , 5

Profit from the Sale of table is Rs. 40 and that from a chair is Rs. 30 determine optimal mix of tables & chairs so as to maximized the total profit Contribution. Let x1 = no. of tables produced and X2 = no. of Chairs produced

Step I. The objective function for maximizing the profit is given by maximize Z=50x1 +30x2 ( objective function )

( Since profit per unit from a table and a chair is Rs. 50 & Rs. 30 respectively). Step II. List down all the constraints. (i) Total time on machine M1 can not exceed 200 hours. 200 4x2 7x1 ≤+∴

( Since it takes 7 hours to produce a table & 4 hours to produce a chair on machine M1) (ii) Total time on machine M2 cannot exceed 400 hours.

200 4x2 7x1 ≤+∴ ( Since it takes 5 hours to produce both a table & a chair on machine M2) Step III Presenting the problem. The given problem can now be formulated as a linear programming model as follows: Maximise Z = 50x1 + 30x2 Subject: : 200 4x 7x 21 ≤+ 400 5x x5 21 ≤+ Further; 0 x x1 2 ≥+ (Since if x1 & x2 < 0 it means that negative quantities of products are being manufactured – which has no meaning).

Example 2.

Alpha Limited produces & sells 2 different products under the brand name black & white. The profit per unit on these products in Rs. 50 & Rs. 40 respectively. Both black & white employ the same manufacturing process which has a fixed total capacity of 50,000 man-hours. As per the estimates of the marketing research department of Alpha Limited, there is a market demand for maximum 8,000 units of Black & 10,000 units of white. Subject to the overall demand, the products can be sold in any possible combination. If it takes 3 hours to produce one unit of black & 2 hours to produce one unit of white, formulate the about as a linear programming model.

Let x1, x2 denote the number of units produced of products black & white respectively.

Step 1: The objective function for maximizing the profit is given by : maximize

Z= 50x1 + 40x2 ( objective function )

Step II: List down all the constraints.

(i) Capacity or man-hours constraint:

( Since it takes 3 hours to produce one unit of x1 & hours to produce 1 unit of x2 & the total available man – hours are 50,000)

(ii) Marketing constraints:

000,81 ≤x

(Since maximum 8,000 units of x1 can be sold )

000,102 ≤x

(Since maximum 10,000 units of x2 can be sold).

Step III: Presenting the problem. Now, the given problem can be written as a linear programming model in the follows:

Maximize Z = 50x1 + 50x2 Subject: : 000,50 2x x3 21 ≤+ 8000 x1 ≤ 000,102 ≤x Further; 0 x x1 2 ≥+( Since if x1,x2< 0, it means that negative Quantities of products are being manufactured – which has no meaning )

Example 3. Good results company manufactures & sells in the export market three different kinds of products P1 , P2 & P3. The anticipated sales for the three products are 100 units of P1, 200 units of P2 & 300 units of P3. As per the terms of the contract Good results must produce at least 50 units of P1 & 70 units of P3. Following is the break – up of the various production times:

Production Hours per Unit Product

Department (A)

Department (B)

Department(C)

Department (D)

(Rs.) Unit

Profit P1 P2 P3

0.05 0.10 0.20

0.06 0.12 0.09

0.07 -- 0.07

0.08 0.30 0.08

15 20 25

Available hours

40.00 45.00 50.00 55.00

Management is free to establish the production schedule subject to the above constraints.

Formulate as a linear programming model assuming profit maximization criterion for Good results company.

Ans. Let X1 , X2 , X3 denote the desired quantities of products P1 , P2 & P3 respectively. Step I. The objective function for maximizing total profits is given by: Maximize Z 15x1 + 20x2 + 25x3 ( objective function ) Step II. List down all the constraints. The available production hours for each of the products must satisfy the following criterion for each department:

(i) 0.05x1+0.10x2+0.20x3 ≤ 40.00 Total hours available on product – wise basis in Department A

(ii) 0.06x1+0.12x2+00.09x3 ≤ 45.00 Total hours available on product – wise basis in Department B

(iii) 0.07x1+0x2+0.07x3 50.00 ≤Total hours available on product – wise basis in Department C

(iv) 0.08x1+0.30x2+0.08x3 ≤ 55.00 Total hours available on product – wise basis in Department D

(v) 10050 1 ≤≤ x Minimum 50 units of P1 must be produced subject of maximum of 100 units.

(vi) 2000 2 ≤≤ xMaximum units of P2 that can be sold is 200 units.

(vii) 30070 3 ≤≤ xMinimum 70 units of P3 must be produced subject to a maximum of 300 units

(viii) Further , x1,x2,x3 0 ≥Since negative values of P1 , P2 & P3 has no meaning Step III: Presenting the Problem The given problem ca be reduced as a LP model as under: Maximise Z=15x1 + 20x2 + 25x3 Subject to: 0.05x1+0.10x2+0.20x3 ≤ 40.00

0.06x1+0.12x2+00.09x3 ≤ 45.00 0.07x1+0x2+0.07x3 ≤ 50.00 0.08x1+0.30x2+0.08x3 ≤ 55.00 10050 1 ≤≤ x 2000 2 ≤≤ x 30070 3 ≤≤ x x1, x2, x3 ≥ 0

Example 4. The management of Surya Chemicals is considering the optimal mix of two possible processes. The values of input & output for both these process are given as follows:

Process Units – inputs Units – Outputs I1 I2 O1 O2

X Y

2 4

6 8

3 5

7 9

Maximum 500 units of Input I1 and 300 units of I2 are available to Surya Chemicals in the local market. The forecasted demand for outputs OI & O2 are at least 5,000 units & 7,000 units respectively. The respective profits from process X & Y are Rs. 1,000 & Rs. 2,000 – per production run. You are required to formulate the above as a linear programming model. Ans. Let X1, X2 represent the number of production runs of process x & y respectively . Step I. The objective function for maximizing the total profits from both the process is given by: Maximise Z 1000x1 + 2000x2 Step II. List down all the constraints (i) 2 5004 21 ≤+ xx Maximum amount of input I1 available for process x & y is 500 units (ii) 6 3008 21 ≤+ xx Maximum amount of input I2 available for process x & y 300 units (iii) 3 000,55 21 ≤+ xx Since market requirement is to produce at least 5000 units of O1 (iv) 7 000,79 21 ≤+ xx Since market requirement is to produce at least 7000 units of O2 Further; x1, x2 ≥ as always. 0 Step III. Presenting the model. Now, the current LP model can be presented as: Maximise Z = 1000x1 + 2000 x2 Subject to: 50042 21 ≤+ xx 6 3008 21 ≤+ xx 3 000,55 21 ≤+ xx 7 000,79 21 ≤+ xx 0, 21 ≥xxExample 5. Chocolate India Ltd. Produces three varieties of Chocolates – Hard, mild & soft from three different inputs I1, I2 & I3 one unit of Hard requires 2 units of I1 and 4 unit of I2. One unit of mild requires 5 units of I1, 4 units of I2 and 3 units of I3 and one unit of soft requires 10 units of I1 & 15 units of I3. The total available of inputs in the company’s warehouse is as under: I1 - 100 units I2 - 400 units I3 - 50 units The profit per unit for hard, mild & soft are Rs. 20, Rs. 30 and Rs. 40 respectively. Formulate the problem so as to maximize the total profit by using linear programming. At the beginning, it is better to present the problem in a tabular form

Inputs Required Product I1 I2 I3

Profit (Rs. Per unit )

HardMildSoft

2 5 10

4 4 --

-- 3 15

20 30 40

Maximum availability of

inputs

100 400 50

Let x1, x2 & x3 denote the no. of units of the three varies of chocolate – Hard, mild & soft respectively. Step 1 The objective function for maximizing the profit is: Maximise Z = 20x1 + 30x2 + 40x3 ( objective function ) Step II List down all the constraints: (i) 2 100105 321 ≤++ xxx (Since Input I1 required for the three products is 2, 5 & 10 units respectively subject to a maximum of 100 units). (ii) 4 40004 321 ≤++ xxx ( Since input I2 required for the three products is 4,4 & 0 units respectively subject to a maximum of 400 units). (iii) 0 50153 321 ≤++ xxx ( Since input I3 required for he three products is 0, 3 & 15 units respectively subject to a maximum of 50 units). Step III. Presenting the problem. The given problem can now be formulated as a linear programming model as follows: Maximise Z = 20x1 + 30x2 + 40x3 Subject to: 2 100105 321 ≤++ xxx 4 40004 321 ≤++ xxx 0 50153 321 ≤++ xxx Further: x 0,, 321 ≥xx

Example 6 Safe & sound Investment Ltd. Wants to invest up to Rs. 10 lakhs into various bonds. The management is currently considering four bonds, the detail on return & maturity of which are as follows:

Bond Type Return Maturity Time α

β

γ

Govt.

Govt. Industrial Industrial

22%

18% 28% 16%

15 years 5 years 20 years 3 years

θ

The company has decided not to put less than half of its investment in the government bonds and that the average age of the portfolio should not be more than 6 years. The investment should be such which maximizes the return on investment, subject to the above restriction.

Formulate the above as a LP problem.

Ans. Let X1 = amount to be invested in bond α Govt. X2 = amount to be invested in bond β Govt. X3 = amount to be invested in bond γ Industrial X4 = amount to be invested in bond θ Industrial Step I. The objective function which maximizes the return on investment Is: Maximise Z = 0.22x1 + 0.18x2 + 0.28x3 + 0.16x4 ( Based on the respective rate of respective rate of return for each bond. Step II. List down all the constraints: (i) Sum of all the investments can not exceed the total fund available. 000,00,104321 ≤+++∴ xxxx

(ii) Sum of investments in government bonds should not be less than 50%. 000,00,521 ≤+∴ xx

(iii) Average :

≤

++++++

riodmaturilypeenotesNumeratord

xxxxxxxx

63205

4321

432115

Or 15 43214321 66663205 xxxxxxxx +++≤+++ Or 6( 0)63()620()65()15 4321 ≤−+−−+ xxxx−

03149 4321 ≤−+−∴ xxxx Step III. Presenting the problem The given problem can now be formulated as a LP model: Maximise Z = 0.22x1 + 0.18x2 + 0.28x3 + 0.16x4 Subject to: 000,00,104321 ≤+++ xxxx 000,00,521 ≤+ xx

9 0314 4321 ≤−+− xxxx Further ; 04321 ≤+++ xxxx

Example7 Good products Ltd. Produces its product in two plants P1 & P2 and distributes this product from two warehouses W1 & W2. Each plant can produce a maximum of 80 units. Warehouse W1 can sell 100 units while W2 can sell only 60 units. Following table shows the cost of shipping one unit from plants to the warehouses:

(cost in Rs) To

from

Warehouse (W1) Warehouse (W2)

Plant (P1) Plant (P1)

40 70

60 75

Determine the total no. of units to be shipped for each plant to each warehouse, such that the capacity of plants is not exceeded, demand at each warehouse is fully satisfied & the total cost of transportation is minimized. Ans. Let Sij = No. of units shipped form plant i to warehouse j. Where i = p1, p2 and j = w1, w2

Further assume C to be the transportation cost of shipping a unit form plant to the warehouse then, we have: Cp1w1 = cost of shipping a unit form plant p1 to warehouse w1, Cp1w2 = cost of shipping a unit form plant p1 to warehouse w2, Cp2w1 = cost of shipping a unit form plant p2 to warehouse w1 & Cp2w2 = cost of shipping a unit form plant p12to warehouse w2, Formulate the problem in a tabular from as under.

P1 to w1 P1 to w2 P2 to w1 P2 to w2 Resources Warehouse W1 demand Warehouse w2 demand

Plant P2 supply Plant p2 supply

shipping costs (Rs.)

1 0 1 0

40

0 1 1 0

60

1 0 0 1

70

0 1 0 1

75

100 units 60 units 80 units 80 units

Formulating as a LP Noble in the usual manner:

Objective – function

Minimize Z = 40 CP1W1 + 60 C P1W2 + 70C P2W1 + 75 Cp2w2 Ware hose constraints. Cp1w1 + Cp1w2 = 100

Cp1w2 + Cp2w2 = 60

Plant constraints

Cp1w1 + Cp1w2 = 80 Cp1w2 + Cp2w2 = 80 Where Cp1w1 , Cp1w2, Cp2w1 , Cp2w2 ≤ 0 (since these denote costs in a minimization problem) Example 8. To maintain good health, a person must fulfil certain minimum daily requirements of several kinds of nutrients. For the sake of simplicity let us assume that only three kinds of these needs to be considered calcium, protein vitamin A. also assume that the person’s diet is to consist of only 2 food items, I & II; whose prices & nutrient content’s are given in the following table. Find out the optimal combination of the two food items that will satisfy the daily requirements & entail the least cost.

Food Calcium Protein Vitamin A Cost of unit FI F2

10 4

5 5

2 6

6 1

Daily minimum Requirement

20

20

12

Ans. Let x1 = Quantity of F1 to be purchased X2 = Quantity of F2 to be purchased Step 1. the objective function for minimizing the total cost is given by: Minimize Z = 6x1 + x2 (objective – function) Step II. List down all the constraints: (i) calcium constraint: 10x1 + 4x2 ≥ 20

(ii) protein constraint: 5x1 + 5x2 ≥ 20 (iii) vitamin constraint: 2x1 + 6x2 ≥ 12 step III. Presenting as a Lp model: minimize Z = 6x2 + X2 10x1 + 4x2 ≥ 20 5x1 + 5x2 ≥ 20 2x1 + 6x2 ≥12 further x1, x2 ≥ 0 Example9. A steel plant manufactures two grades of steel S1 & S2 . Data given below shows the total resources consumed & profit per unit associated with S1 & S2 .iron and labour are the only resources which are consumed in the manufacturing process. The

manager of the firm wishes to determine the different units of S1 & S2 . which should be manufactured to maximize the total profit. Resource utilized Unit-requirement

S1 S2

Amount

Available Lron (kg) labour

(Hours) 30 5

20 10

300 110

Profit (Rs.) 6 8

Ans. Let x1 = no. of units of S1 to be manufactured. x2 = no. of units of S2 to be manufactured. Step I The objective function for maximizing to be manufactured. Z = 6x1 + 8x2 ( objective – function ) ( Since profit per unit of S1 & S2 are Rs. 6 and Rs. 8 respectively ) Step II. List down all the constraints: (i) Iron – constraint: 30x1 +20x2 300 ≤(ii) Labour – constraint:

5x1 +10x2 110 ≤Step III. Presenting as a LP Model Maximize Z = 6x1 + 8x2 Subject to: 30x1 +20x2 300 ≤ 5x1 +10x2 ≤ 110 Further : x1,x2≥ 0 Example 10. Mr. Khanna is exploring the various investment options for maximizing his return on investment. The investments that can be made by him pertain to the following areas: Govt. bonds, fixed deposits of companies equity shares time deposits in banks, Indira Vikas Patra & real estate. The data pertaining to the return on investment, the number of years for which the funds will be blocked to earn this return on investment and the related risks are as follows: Option Return No. of years Risk factor

Govt. bonds Company deposits

Equity shares

6% 15% 20%

15 3 6

1 3 7

Time deposits Indira Vikas Patra

Real Estate

10% 12% 25%

3 6 10

1 1 2

The average risk required by Mr. Khanna should not exceed 3 and the funds should not be fixed for more than 20 years. Further he would necessarily Formulate the above as a LP model. Ans. Following the familiar methods of Linear programming, the objective function is: Maximize 0Z = 654321 25.012.010.020.015.006. xxxxxx +++++ ( Maximizing total return on investment ) Subject to the following constraint equations: 2010636315 654321 ≤+++++ xxxxxx ( Funds )20 yearsthanmoreforfixedbenotshould∴ 3 (273 4321 ≤++++ xxxx ∴ Risk required at most = 3)

40.06 ≥x (∴ Investment in real estate is at least 40%) Further 6....2,10 =≥ iforxi i.e., 0,,,,, 654321 ≥xxxxxxwhere x1 = Percentage of total funds to be invested in Govt. – bonds. x2 = Percentage of total funds to be invested in company deposits. x3 = Percentage of total funds to be invested in equity shares. x4 = Percentage of total funds to be invested in time deposits. x5 = Percentage of total funds to be invested in indira vikas patra x6 = Percentage of total funds to be invested in real estate

Example 11. Round –the –clock Ltd., a departmental store has the following daily requirement for staff.

Period Time ( 24 hrs a day) Minimum staff requirement

1 2 3 4 5 6

6 am -- 10 am 10 am – 2 pm 2 pm – 6 pm 6 pm – 10 pm 10 pm – 2 am 2 am – 6 am

12 17 25 18 20 16

Staff reports to the store at the start of each period & works for 8 consecutive hours. The departmental store wants to find out the minimal number of staff to be employed so that there will be sufficient number of employees available for each period.

You are required to formulate the problem as a linear programming model giving clearly the constraints & the objective function. Solution be the number of staff members required to be present in the store for the corresponding 1 to 6 periods.

654321 ,,,,, xxxxxx

Proceeding in the usual manner:

The objective function can be expressed as under: Minimize xZ = 654321 ,,,,, xxxxx ( Since objective is to minimize the no. of staff) Constraint equation: Since each person has to work for 8 consecutive hours, the X1 employee who are employed during the Ist period shall still be on duty when the IInd period starts. Thus, during the II period there will be x1 + x2 employees.

Since the minimal number of staff required during the 2nd period is given to be 17, therefore we must have the first constraint equation as: x1 +x2 ≥ 17 like wise the remaining constraint equations can be written as : x2 +x3 ≥ 25 x3 +x4 ≥ 18 x4 +x5 ≥ 20 x5 +x6 ≥ 16 and x16+x1 ≥ 2 where X1, X2 ……….. ≥ 0 Example12.The management of Quality Toys wants to determine the number of advertisement to be placed in three monthly magazines M1 , M2 & M3 the primary objective for advertising is to maximize the total exposure to the customers & the potential buyers of its high quality & safe toys. Percentages of readers for each of the 3 magazines M1 , M2 & M3 are know with the help of a readership survey. The following information is provided. Magazines Readers (in lakh) 2 1.60 1.40 Principal buyer 10% 10% 5% Cost / advertisement(Rs.)

5000 4000 3000

The maximum advertisement budget is Rs. 10 lakhs. The management has already decided that magazine M1 should not have more than 12 advertisements & that M2 & M3 can each have at least 2 advertisements. Formulate the above as a LP model. (exposure in a magazine = No. of advertisement placed X no. of principal buyers) Ans. Let X1 , X2 , X3 denoted the required no. of advertisements in magazines M1 , M2 & M3 respectively. Step 1. the total exposure of principal buyers of the magazine is: Z = (10% of 200000) X1 +(10% of 160000) X2 + (5% of 140000) X3 = 20000 X1 + 16000 X2 + 7000 X3 Step 2. list down the constrain equations: X1 ≤ 12 9since M1 can not have more than 12 advertisements) X2 ≥ 2 since M1 M2 can have at least 2 advertisements X3 ≥ 2

Step 3. present as LP model i.e., Z = 20000 X1 + 16000 X2 +7000 X3 Maximize (Exposures are ot be maximized) Subject to : X1≤ 12 X2≤ 2 X3≤ 2 Further X1 , X2 , X3 ≥ 0 Example 13.Precise manufacturing works produces a product each unit of which consists of 10 units of component A & 8 unit of component. B Both these components are manufactured form two different raw materials of which 100 units & 80 unit are available respectively. The manufacture of both these components A & B is done in three distinct departments. The following results pertaining to both the components are given: Department Input per cycle Output per cycle Raw material Raw material Component Component X Y A B P 18 16 17 15 Q 15 19 16 19 R 13 18 18 14 Formulate the given problem as a linear programming model so as to find out the optimal m=number of production runs for each of the three departments that will maximize the total number of complete units of the final product. Let X1 , X2 , X3 respect the no. of production runs for the three departments P,Q,& R respectively Step 1. Write the objective function

The objective is to maximize the total no. of units of final product. Since each unit of product requires 10 units of component A& 8 units of component B, there fore the maximum number of units of the final products can not exceed the smaller value of : (17 X1 + 16X2 + 18 X3 ) /10 (i.e. the np. Of units of component A produced by different departments divided by the no. units of A required for the final product ) also 15X1 + 19 X2 + 14X3 / 8 (i.e. the no of units of component B manufactured by various departments divided by the no. of units of B required for the final product) the objective function hence becomes: Maximize

Z = Minimum of

++++

8 X 14 19X X 15,

10 X 18 16X X 17 32 132 1

Step II. Give the constraint equations: Raw material constraints:

100131518 321 ≤++ xxx ( Raw material : X)

80181916 321 ≤+