INTERNATIONAL SCHOOL OF BUSINESS AND TECHNOLOGY

COURSE WORK

OPERATIONS RESEARCH

SUBJECT CODE: MB0032

NAME : TWAHIRWA TADEO

ROLL NO : 540811570

COURSE : MBA

SEMESTER: 2

SIKKIM MANIPAL UNIVERSITY (INDIA)UGANDA LEARNING CENTER

QN.1. Describe in details the different scopes of application of Operations Research.

ANS:

Where ever there is any problem, simple or complicated, operation research techniques may be

applied to find the best solution. Examples of applications in which operations research is

currently used include:

Operation research is used in defense operations. Right from the beginning of World War two,

operation research has widely been used in defense systems. Through various departments in

modern armies like air force, navy and army, in various activities of administration, intelligence,

training and supply, operation research techniques are now widely employed. The applications of

modern war fare techniques in each of the components of military organizations require expertise

knowledge in respective fields. Each of these components works to drive maximum gains from

its operation and there is always a possibility that the strategy beneficial to one component may

have an adverse effect on the other. For example, Operations research doubled the success rate of

aerial attacks on submarines by recommending a shallower detonation setting on the depth

charges being dropped by aircraft, it also doubled the on-target bomb rate of B-29s bombing

Japan from the Marianas Islands by increasing the training ratio from 4 to 10 percent of flying

hours.

Operation research is used in industry. The modern systems in production are so complex that

the optimum point of operation in its various components cannot be intuitively judged by an

individual. The business environment is ever changing and any decision useful at one time may

not be so good a time later. There is therefore, need to always check and validate decisions

continually, against the situation. Due to division of labor, each department must ensure

efficiency and this can only be achieved by employing operation research techniques. For

example, the production department ensures maximum out put with minimum in put, whilst the

marketing department ensures maximum sales returns while minimizing the cost per unit sale.

Operation research helps in designing the layout of a factory for efficient flow of materials.

Operation research is also employed during the planning process. In modern times it has become

necessary for every government to have careful planning, for economic development of the

country. Operation research techniques can be fruitfully applied to maximize the per capita

income, with minimum sacrifice and time. A government can thus use various operation research

techniques for framing future economic and social policies. This is particularly true with

governments which use extensively evidence-based policy.

In agriculture, various operation research techniques are also employed. Due to population

explosion, there is need to maximize agricultural out put to cater for the ever increasing

population. But this can only be achieved if the courses of action taken take into account various

restrictions prescribed in order to ensure improved productivity.

In hospitals operation research is equally relevant. The various methods here are used to solve

waiting problems in out-patient departments of big hospitals. The administrative problems of

hospital organization can also be solved by operation research techniques.

In transport, operation research is widely employed. Different techniques are applied to regulate

the arrival of trains and processing times, minimize the passengers waiting time and reduce

congestion, formulate suitable transportation policy, reducing the costs and time of trans-

shipment. Other examples include; road traffic management and one way street allocations i.e.

allocation problems, determining the routes of school buses (or city buses) so that as few buses

are needed as possible.

In research and development, operation research helps in control of projects, production

planning, identifying those processes in a complex project which affect the overall duration of

the project and blending of raw materials in oil refineries determining optimal prices, in many

retail and B2B settings, within the disciplines of pricing science.

It is also employed in constructing of telecommunications network at low cost while still

guaranteeing quality of service or Quality of Experience if particular connections become very

busy or get damaged in the connection process.

It helps in globalizing operations processes in order to take advantage of cheaper materials,

labor, land or other productivity inputs.

QN.2. what do you understand by Linear Programming Problem? What are the

requirements of L.P.P.? What are the basic assumptions of L.P.P.?

ANS:

Linear programming problem is a class of mathematical programming in which the functions

representing the objective and the constraints are linear. Here by optimization, we mean either to

maximize or minimize the objective functions.

The requirements of L.P.P include;

Decisions variables and their relationship

Well defined objective function

Existence of alternative courses of action

Non-negative conditions on decision variables

The assumptions of Linear Programming include the following;

• Proportionality. Given a variable xj, its contribution to the objective is cjxj and its

contribution to i th constraint is aijxj.

• Additivity. Guarantees that the total cost is the sum of individual costs.

• Divisibility. Ensures that decision variables can be divided into fractions

• Deterministic. Coefficients cj, aij, bi are all know deterministically.

QN.3.Describe the different steps needed to solve a problem by simplex method.

ANS:

The Simplex Method has its own deficiencies. For example, it requires that all variables be non-

negative ( 0); also, all other constraints must be in form with non-negative right-hand-side

(RHS) values. The steps needed to solve a problem by simplex include the following;

1. Convert the LP to the following form:

Convert the minimization problem into a maximization one (by multiplying the

objective function by (-1).

All variables must be non-negative.

All RHS values must be non-negative (multiply both sides by -1, if needed).

All constraints must be in form (except the non-negativity conditions). No strictly

equality or constraints are allowed.

If this condition cannot be satisfied, then use the Initialization of the Simplex

Method: Artificial-Free.

2. Convert all constraints to equalities by adding a different slack variable for each

one of them.

3. Construct the initial simplex tableau with all slack variables in the BVS. The last

row in the table contains the coefficient of the objective function (row Cj).

4. Determine whether the current tableau is optimal. That is:

If all RHS values are non-negative (called, the feasibility condition)

If all elements of the last row, that is Cj row, are non-positive (called, the

optimality condition).

If the answers to both of these two questions are yes, then stop. The current

tableau contains an optimal solution. Otherwise, go to the next step.

5. If the current BVS is not optimal, determine, which non basic variable should

become a basic variable and, which basic variable should become a non basic

variable. To find the new BVS with the better objective function value, perform

the following tasks:

o Identify the entering variable: The entering variable is the one with the

largest positive Cj value (In case of a tie, we select the variable that

corresponds to the leftmost of the columns).

o Identify the outgoing variable: The outgoing variable is the one with

smallest non-negative column ratio (to find the column ratios, divide the

RHS column by the entering variable column, wherever possible). In case

of a tie we select the variable that corresponds to the upmost of the tied

rows.

QN.4. Describe the economic importance of the Duality concept.

ANS:

Duality is a resource allocation model in which the objective is to maximize revenue or profit

subject to limited resources. Looking at the problem from a duality point of view, the associated

dual problem offers interesting economic interpretation of the L.P resource allocation model.

QN.5.How can you use the Matrix Minimum method to find the initial basic feasible

solution in the transportation problem.

ANS:

This follows the following steps:

1. Determine the smallest cost in the cost matrix of the transportation table. Let it be Cij ,

allocate Xij = min(ai,bj) in the cell (i,j)

2. If Xij = ai cross off the ith row off the transportation table and decrease bj by ai continue

to the next step. If Xij = bj cross off the ith column of the transportation table and

decrease ai by bj go to the next step.

3. Repeat steps 1 and 2 for the resulting reduced transportation table until all the rim

requirements are satisfied whenever the minimum cost is not unique make an arbitrary

among the minima.

Consider the example below:

D1 D2 D3 D4 SUPPLY

O1 1 2 3 4 6O2 4 3 2 0 8O3 0 2 2 1 10

DEMAND 4 6 8 6 24

1*4+2*2+2*4+2*8+1*6=38 being the minimum transportation cost

QN.6 Describe the Integer Programming Problem. Describe the Gomory’s All-I.P.P.

method for solving the I.P.P. problem.

ANS:

When formulating LP's we often find that, strictly, certain variables should be regarded as taking

integer values but, for the sake of convenience, we let them take fractional values reasoning that

the variables were likely to be so large that any fractional part could be neglected. Whilst this is

acceptable in some situations, in many cases it is not, and in such cases we must find a numeric

solution in which the variables take integer values.

Problems in which this is the case are called integer programs (IP's) and the subject of solving

such programs is called integer programming (also referred to by the initials IP).

IP's occur frequently because many decisions are essentially discrete (such as yes/no, go/no-go)

in that one (or more) options must be chosen from a finite set of alternatives.

Note here that problems in which some variables can take only integer values and some variables

can take fractional values are called mixed-integer programs (MIP's).

As for formulating LP's the key to formulating IP's is practice. Although there are a number of

standard tricks available to cope with situations that often arise in formulating IP's it is probably

true to say that formulating IP's is a much harder task than formulating LP's.

Gomory’s All – IPP Method.

An optimum solution to an I.P.P. is first obtained by using simplex method ignoring the

restrictions of integral values. In the optimum solution if all the variables have integer values, the

current solution will be the desired optimum integer solution. Otherwise the given IPP is

modified by inserting a new constraint called Gomory’s or secondary constraint which represents

necessary condition for integrability and eliminates some non integer solution without losing any

integer solution.

After adding the secondary constraint, the problem is then solved by dual simplex method to get

an optimum integral solution. If all the values of the variables in this solution are integers, an

optimum inter-solution is obtained, other wise another new constraint is added to the modified

LPP and the procedure is repeated.

An optimum integer solution will be reached eventually after introducing enough new constraints

to eliminate all the superior non integer solutions. The construction of additional constraints,

called secondary or Gomory’s constraints is so very important that it needs special attention.