ASSIGNMENTS

ASSIGNMENTS

MB 0032

(2 credits)

Set 1

Marks 30

Operations ResearchNote: Each Question carries 10 marks

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

Answer: Operations Research (OR) in the USA, South Africa and Australia, and Operational

Research in Europe and Canada, is an interdisciplinary branch of applied mathematics and formal

science that uses methods such as mathematical modeling, statistics, and algorithms to arrive at

optimal or near optimal solutions to complex problems. It is typically concerned with optimizing

the maxima (profit, assembly line performance, crop yield, bandwidth, etc) or minima (loss, risk,

etc.) of some objective function. Operations research helps management achieve its goals using

scientific methods. The terms operations research and management science are often used

synonymously.

When a distinction is drawn, management science generally implies a closer relationship to the

problems of business management. The field of operations research is closely related to Industrial

engineering. Industrial engineers typically consider Operations Research (OR) techniques to be a

major part of their toolset. Some of the primary tools used by operations researchers are statistics,

optimization, probability theory, queuing theory, game theory, graph theory, decision analysis,

and simulation. Because of the computational nature of these fields, OR also has ties to computer

science, and operations researchers use custom-written and off-the-shelf software.

Operations research is distinguished by its frequent use to examine an entire management

information system, rather than concentrating only on specific elements (though this is often done

as well). An operations researcher faced with a new problem is expected to determine which

techniques are most appropriate given the nature of the system, the goals for improvement, and

constraints on time and computing power. For this and other reasons, the human element of OR is

vital. Like any other tools, OR techniques cannot solve problems by themselves.

Scope of operation Research

Examples of applications in which operations research is currently used include:

Critical path analysis or project planning: identifying those processes in a complex project which

affect the overall duration of the project

1. Designing the layout of a factory for efficient flow of materials

2. constructing a telecommunications network at low cost while still guaranteeing QoS

(quality of service) or QoS (Quality of Experience) if particular connections become very

busy or get damaged

3. Road traffic management and 'one way' street allocations i.e. allocation problems.

4. Determining the routes of school buses (or city buses) so that as few buses are needed as

possible

5. designing the layout of a computer chip to reduce manufacturing time (therefore reducing

cost)

6. Managing the flow of raw materials and products in a supply chain based on uncertain

demand for the finished products

7. Efficient messaging and customer response tactics

8. Robotizing or automating human-driven operations processes

9. Globalizing operations processes in order to take advantage of cheaper materials, labor,

land or other productivity inputs

10. Managing freight transportation and delivery systems (Examples: LTL Shipping,

intermodal freight transport)

11. Scheduling:

Personnel staffing

Manufacturing steps

Project tasks

Network data traffic: these are known as queuing models or queueing systems.

sports events and their television coverage

12. blending of raw materials in oil refineries

13. determining optimal prices, in many retail and B2B settings, within the disciplines of

pricing science

Operations research is also used extensively in government where evidence-based policy is used.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.?

Answer: Linear programming problem (LPP):

The standard form of the linear programming problem is used to develop the procedure for

solving a general programming problem.

A general LPP is of the form

Max (or min) Z = c1x1 + c2x2 + +cnxn

x1, x2, ....xn are called decision variable. subject to the constraints

c1, c2,. Cn, a11, a12,. amn are all known constants

Z is called the "objective function" of the LPP of n variables which is to be maximized or

minimized.

Requirements of L.P.P :

There are mainly four steps in the mathematical formulation of linear programming problem as a

mathematical model. We will discuss formulation of those problems which involve only two

variables.

Identify the decision variables and assign symbols x and y to them. These decision

variables are those quantities whose values we wish to determine.

Identify the set of constraints and express them as linear equations/inequations in terms of

the decision variables. These constraints are the given conditions.

Identify the objective function and express it as a linear function of decision variables. It

might take the form of maximizing profit or production or minimizing cost.

Add the non-negativity restrictions on the decision variables, as in the physical problems,

negative values of decision variables have no valid interpretation.

There are many real life situations where an LPP may be formulated. The following examples

will help to explain the mathematical formulation of an LPP.

Example-1. A diet is to contain at least 4000 units of carbohydrates, 500 units of fat and 300

units of protein. Two foods A and B are available. Food A costs 2 dollars per unit and food B

costs 4 dollars per unit. A unit of food A contains 10 units of carbohydrates, 20 units of fat and 15

units of protein. A unit of food B contains 25 units of carbohydrates, 10 units of fat and 20 units

of protein. Formulate the problem as an LPP so as to find the minimum cost for a diet that

consists of a mixture of these two foods and also meets the minimum requirements.

Suggested answer:

The above information can be represented as

Let the diet contain x units of A and y units of B.

Total cost = 2x + 4y

The LPP formulated for the given diet problem is

Minimize Z = 2x + 4y subject to the constraints

Basic Assumptions of L.P.P:

Linear programming is applicable only to problems where the constraints and objective function

are linear i.e., where they can be expressed as equations which represent straight lines. In real life

situations, when constraints or objective functions are not linear, this technique cannot be used.

Factors such as uncertainty, weather conditions etc. are not taken into consideration.

There may not be an integer as the solution, e.g., the number of men required may be a

fraction and the nearest integer may not be the optimal solution.

i.e., Linear programming techniques may give practical valued answer which is not desirable.

Only one single objective is dealt with while in real life situations, problems come with

multi-objectives.

Parameters are assumed to be constants but in reality they may not be so.3. Describe the different steps needed to solve a problem by simplex method.

Answer: Simplex method

The simplex method is a method for solving problems in linear programming. This method,

invented by George Dantzig in 1947, tests adjacent vertices of the feasible set (which is a

polytope) in sequence so that at each new vertex the objective function improves or is unchanged.

The simplex method is very efficient in practice, generally taking 2m to 3m iterations at most

(where m is the number of equality constraints), and converging in expected polynomial time for

certain distributions of random inputs (Nocedal and Wright 1999, Forsgren 2002). However, its

worst-case complexity is exponential, as can be demonstrated with carefully constructed

examples (Klee and Minty 1972). A different type of methods for linear programming problems

are interior point methods, whose complexity is polynomial for both average and worst case.

These methods construct a sequence of strictly feasible points (i.e., lying in the interior of the

polytope but never on its boundary) that converges to the solution. Research on interior point

methods was spurred by a paper from Karmarkar (1984). In practice, one of the best interior-point

methods is the predictor-corrector method of Mehrotra (1992), which is competitive with the

simplex method, particularly for large-scale problems.

Dantzig's simplex method should not be confused with the downhill simplex method (Spendley

1962, Nelder and Mead 1965, Press et al. 1992). The latter method solves an unconstrained

minimization problem in n dimensions by maintaining at each iteration n+1 points that define a

simplex. At each iteration, this simplex is updated by applying certain transformations to it so that

it "rolls downhill" until it finds a minimum.

The Simplex Method is "a systematic procedure for generating and testing candidate vertex

solutions to a linear program." (Gill, Murray, and Wright, p. 337) It begins at an arbitrary corner

of the solution set. At each iteration, the Simplex Method selects the variable that will produce

the largest change towards the minimum (or maximum) solution. That variable replaces one of its

compatriots that is most severely restricting it, thus moving the Simplex Method to a different

corner of the solution set and closer to the final solution. In addition, the Simplex Method can

determine if no solution actually exists. Note that the algorithm is greedy since it selects the best

choice at each iteration without needing information from previous or future iterations.

The Simplex Method solves a linear program of the form described in Figure 3. Here, the

coefficients cj represent the respective weights, or costs, of the variables xi. The minimized

statement is similarly called the cost of the solution. The coefficients of the system of equations

are represented by aij, and any constant values in the system of equations are combined on the

right-hand side of the inequality in the variables bi.

Combined, these statements represent a linear program, to which we seek a solution of minimum

cost.

Solving this linear program involves solutions of the set of equations. If no solution to the set of

equations is yet known, slack variables x n+1, x n+2, ., x n+m, adding no cost to the solution,

are introduced. The initial basic feasible solution (BFS) will be the solution of the linear program

where the following holds:

Once a solution to the linear program has been found, successive improvements are made to the

solution. In particular, one of the nonbasic variables (with a value of zero) is chosen to be

increased so that the value of the cost function, , decreases. That variable is then

increased, maintaining the equality of all the equations while keeping the other nonbasic variables

at zero, until one of the basic (nonzero) variables is reduced to zero and thus removed from the

basis. At this point, a new solution has been determined at a different corner of the solution set.

The process is then repeated with a new variable becoming basic as another becomes nonbasic.

Eventually, one of three things will happen. First, a solution may occur where no nonbasic

variable will decrease the cost, in which case the current solution is the optimal solution. Second,

a non-basic variable might increase to infinity without causing a basic variable to become zero,

resulting in an unbounded solution. Finally, no solution may actually exist and the Simplex

Method must abort. As is common for research in linear programming, the possibility that the

Simplex Method might return to a previously visited corner will not be considered here.

The primary data structure used by the Simplex Method is "sometimes called a dictionary, since

the values of the basic variables may be computed (looked up) by choosing values for the

nonbasic variables." (Gill, Murray, and Wright, p. 337) Dictionaries contain a representation of

the set of equations appropriately adjusted to the current basis. The use of dictionaries provide an

intuitive understanding of why each variable enters and leaves the basis. The drawback to

dictionaries, however, is the necessary step of updating them which can be time-consuming.

Computer implementation is possible, but a version of the Simplex Method has evolved with a

more efficient matrix-oriented approach to the same problem. This new implementation became

known as the Revised Simplex Method.The steps of the Simplex Method also need to be expressed in the matrix format of the Revised

Simplex Method. The basis matrix, B, consists of the column entries of A corresponding to the

coefficients of the variables currently in the basis. That is if x2 is the fourth entry of the basis,

then [ a12 a22 am2]T is the fourth column of B. (Note that B is therefore an m X m matrix.)

The non-basic columns of A constitute a similar though likely not square, matrix referred to here

as V.1. Describe the broad classification of Operations Research models in details. Name the

different steps needed in OR approach of problem solving?

Answer: A model is known as the representation of the reality. It is known as an idealized

representation or abstraction of a real life system. The main objective of this model is to identify

significant factors and their interrelationship. A model is helpful is decision making as it provides

a simplified description of complexities and uncertainties of a problem in logical structure.

A broad classification of OR models:

a) Physical modes include all form of diagrams, graphs and charts. They are designed to deal with

specific problems. They bring out significant factors and inter-relationship in pictorial firm so as

to facilitate analysis.

There are two types:

1) Ieonic models and 2) Analog models

Iconic model is known as an image of an object or system that is represented on a small scale. We

can say that these models can simulate the actual performance of a product.

On the other hand analog models are small physical systems that have similar characteristics and

work like an object. For example- Toy.

b) Mathematical Model or Symbolic models represent the decision variable of the system. The

model employs a set of mathematical symbols also. The variables are related by mathematical

system also. For example - Allocation, sequencing, replacement models etc.

c) It is by nature of Environment

We have 1) Deterministic model in which every thing is defined and the results are certain. Eg:

EOQ model 2) Probabilistic models in which the input and output variables follow a probability

distribution Eg: Games Theory.

d) By the extent of Generality: The tow models belonging to this class are 1) General model can

be applied in general and does not pertain to one problem only. Eg: Linear Programming 2)

Specific model is applicable under specific condition only. For example - Sales can response

curve or equation which can be known as a function of advertising that is applicable in the

marketing function alone.

The scientific method translates a real given problem into a mathematical representation which is

solved and retransformed into the original context. The OR approach to problem solving consists

of the following steps:

1) Definition of the problem

The first and the most important requirement is that the root problem should be identified and

understood. The problem should be identified properly, this indicates three major aspects:

1) A description of the goal or the objective of the study, 2) an identification of the decision

alternative to the system, and 3) a recognition of the limitations, restrictions and requirements of

the system.

2) Construction of the model

Depending on the definition of the problem, the operations research team should decide on the

most suitable model for representing the system. Such a model should specify quantitative

expressions for the objective and the constraints of the problem in terms of its decision variables.

3) Solution of the model

Once an appropriate model has been formulated the next stage in the analysis calls for its solution

and the interpretation of the solution in the context of the given problem A solution to a model

implies determination of a specific set of decision variables that would yield on optimum

solution. An optimum solution is one which maximizes or minimizes the performance of any

measure in a model subject to the condition and constraints imposed on the model.

4) Validation the model

A model is a good representative of a system, and then the optimal solution must improve the

systems performance. A common method for testing the validity of a model is to compare its

performance with some post data available for the actual system.

5) About Implementation of the final result

The optimal solution obtained from a model should be applied practice to improve the

performance of the system and the validity of the solution should be verified under changing

conditions.2. Describe the graphical method to solve the Linear Programming Problem. Use the

following example

Maximize Z = 30x1 + 40x2

Subject to the constraints

1.5x1 + 1.9x2 600

0.3 x1 + 0.2x2 100

0.0x1 + 0.2x2 30

and x1 0, x2 0

Answer: A LPP with 2 decision variables x1 and x2 can be solved easily by graphical method.

We consider the x1 x2 plane where we plot the solution space, which is the space enclosed by

the constraints. Usually the solution space is a convex set which is bounded by a polygon since a

linear function attains extreme (maximum or minimum) values only on boundary of the region, it

is sufficient to consider the vertices of the polygon and find the value of the objective function in

these vertices. By comparing the vertices of the objective function at these vertices, we obtain the

optimal solution of the problem.

The method of solving a LPP on the basis of the above analysis is known as the graphical

method. The working rule for the method is as follows:

Working Rule:

Step I: Write down the equations by replacing the inequality symbols by the equality symbol in

the given constraints.

Step II: Plot the straight lines represented by the equations obtained in step I.

Step III: Identify the convex polygon region relevant to the problem. We must decide on which

side of the line, the halfplane is located.

Step IV: Determine the vertices of the polygon and find the values of the given objective

function Z at each of these vertices. Identify the greatest and least of these values. These are

respectively the maximum and minimum value of Z.

Step V: Identify the values of (x1, x2) which correspond to the desired extreme value of Z. This

is an optimal solution of the problem.

Solution by Graphical Method

Let the horizontal axis represent x1and vertical axis represent x2. plot the constraint lines,

feasibility region has been shown in fig.

Any point on the thick line or inside the shaded portion will satisfy all the restrictions of the

problem then ABCDE is the feasibility region carried out by the constraints operating on the

objective function. This depicts the limits within which the values of the decision variables are

permissible. The intersection points C and D can be solved by the linear equations 0.1x2 < 30,

0.1x1 + 1.5x2 = 600, and 0.2x1 + 0.2x2 < 100 i.e. C(150,300) and D(300,180).

We work out the revenues at different corners points as tabulated below

At Point Feasible Solution of the

product mix

Corresponding revenue Total

revenue

X1 X2 From X1 From X2

A

B

C

D

E

0

0

15

0

30

0

50

0

0

30

0

30

0

18

0

0

0

0

7500

1500

0

2500

0

0

2400

2400

0

1440

0

0

0

24000

31500

29400

25000

From the above table we find that revenue is maximum at Rs.31500 when 150 unit of X1 and 300

units of X2 are produced.3. Describe the Penalty Cost method (Big M Method) for solving L.P.P.

Answer: Penalty Cost Method Or Big-M Method

Consider a L.P.P. when atleast one of the constraints is of the type > or = . While expressing in

the standard form, add a non negative variable to each of such constraints. These variables are

called artificial variables. Their addition causes violation of the corresponding constraints, since

they are added to only one side of an equation, the new system is equivalent to the old system of

constraints if and only if the artificial variables are zero. To guarantee such assignments in the

optimal solution, artificial variables are incorporated into the objective function with large

positive coefficients in a minimization program or very large negative coefficients in a

maximization program. These coefficients are denoted by } M.

Whenever artificial variables are part of the initial solution X0, the last row of simplex table will

contain the penalty cost M. The following modifications are made in the simplex method to

minimize the error of incorporating the penalty cost in the objective function. This method is

called Big M method or Penalty cost method.

1) The last row of the simplex table is decomposed into two rows, the first of which involves

those terms not containing M, while the second involves those containing M.

2) The Step 1 of the simplex method is applied to the last row created in the above modification

and followed by steps 2, 3 and 4 until this row contains no negative elements. Then step 1 of

simplex algorithm is applied to those elements next to the last row that are positioned over zero in

the last row.

3) Whenever an artificial variable ceases to be basic, it is removed from the first column of the

table as a result of step 4, it is also deleted from the top row of the table as is the entire column

under it.

4) The last row is removed from the table whenever it contains all zeroes.

5) If non zero artificial variables are present in the final basic set, then the program has no

solution.

In contrast, zero valued artificial variables in the final solution may exist when one or more of the

original constraint equations are redundant.

4. Why Duality concept is important in OR? Describe the economic importance of the

Duality concept.

Answer: The Importance of Duality Concept Is Due To Two Main Reasons

i. If the primal contains a large number of constraints and a smaller number of variables, the

labour of computation can be considerably reduced by converting it into the dual problem and

then solving it.

ii. The interpretation of the dual variable from the loss or economic point of view proves

extremely useful in making future decisions in the activities being programmed.

Economic importance of duality concept

The linear programming problem can be thought of as a resource allocation model in which the

objective is to maximize revenue or profit subject to limited resources. Looking at the problem

from this point of view, the associated dual problem offers interesting economic interpretations of

the L.P resource allocation model. We consider here a representation of the general primal and

dual problems in which the primal takes the role of a resource allocation model.

Primal

Maximize

From the above resource allocation model, the primal problem has n economic activities and m

resources. The coefficient cj in the primal represents the profit per unit of activity j.

Resource i, whose maximum availability is bi, is consumed at the rate aij units per unit of activity

j.

Interpretation of Duel Variables

For any pair of feasible primal and dual solutions,

(Objective value in the maximization problem) (Objective value in the minimization problem)

At the optimum, the relationship holds as a strict equation. Note: Here the sense of optimization is

very important. Hence clearly for any two primal and dual feasible solutions, the values of the

objective functions, when finite, must satisfy the following inequality.

The strict equality, z = w, holds when both the primal and dual solutions are optimal.

Consider the optimal condition z = w first given that the primal problem represents a resource

allocation model, we can think of z as representing profit in Rupees. Because bi represents the

number of units available of resource i, the equation z = w can be expressed as profit (Rs) =

(units of resource i) x (profit per unit of resource i) This means that the dual variables yi,

represent the worth per unit of resource i [variables yi are also called as dual prices, shadow

prices and simplex multipliers]. With the same logic, the inequality z < w associated with any two

feasible primal and dual solutions is interpreted as (profit) < (worth of resources) This

relationship implies that as long as the total return from all the activities is less than the worth of

the resources, the corresponding primal and dual solutions are not optimal. Optimality is reached

only when the resources have been exploited completely, which can happen only when the input

equals the output (profit). Economically the system is said to remain unstable (non optimal) when

the input (worth of the resources) exceeds the output (return). Stability occurs only when the two

quantities are equal.5. Describe the Matrix Minimum method of finding the initial basic feasible solution in the

transportation problem.

Answer: The Initial basic Feasible solution using Matrix Minimum Method

Let us consider a T.P involving m-origins and n-destinations. Since the sum of origin capacities

equals the sum of destination requirements, a feasible solution always exists.

Any feasible solution satisfying m + n 1 of the m + n constraints is a redundant one and hence

can be deleted. This also means that a feasible solution to a T.P can have at the most only m + n

1 strictly positive component, otherwise the solution will degenerate.

It is always possible to assign an initial feasible solution to a T.P. in such a manner that the rim

requirements are satisfied.

This can be achieved either by inspection or by following some simple rules. We begin by

imagining that the transportation table is blank i.e. initially all xij = 0. The simplest procedures

for initial allocation discussed in the following section.

Matrix Minimum Method

Step 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)

Step 2: If xij = ai cross off the ith row of the transportation table and decrease bj by ai go to step

3.

if xij = bj cross off the ith column of the transportation table and decrease ai by bj go to step 3.

if xij = ai= bj cross off either the ith row or the ith column but not both.

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 choice

among the minima.

6. What do you understand by the Integer Programming Problem? Describe the Gomorys

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

Answer: Integer Programming Problem

The Integer Programming Problem I P P is a special case of L P P where all or some variables are

constrained to assume nonnegative integer values. This type of problem has lot of applications in

business and industry where quite often discrete nature of the variables is involved in many

decision making situations. Eg. In manufacturing the production is frequently scheduled in terms

of batches, lots or runs In distribution, a shipment must involve a discrete number of trucks or

aircrafts or freight cars .

An integer programming problem can be described as follows:

Determine the value of unknowns x1, x2, , xn so as to optimize z = c1x1 +c2x2 + . . .+ cnxn

subject to the constraints ai1 x1 + ai2 x2 + . . . + ain xn =bi , i = 1,2,,m and xj > 0 j = 1, 2,

,n where xj being an integral value for j = 1, 2, , k n.

If all the variables are constrained to take only integral value i.e. k = n, it is called an all (or pure)

integer programming problem. In case only some of the variables are restricted to take integral

value and rest (n k) variables are free to take any non negative values, then the problem is

known as mixed integer programming problem.

Gomorys All IPP Method

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

restriction 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 Gomorys or secondary constraint which represents

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

integral 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, otherwise another new constrained is

added to the modified L P P 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 Gomorys

constraints, is so very important that it needs special attention.

The iterative procedure for the solution of an all integer programming problem is as follows:

Step 1: Convert the minimization I.P.P. into that of maximization, if it is in the minimization

form. The integrality condition should be ignored.

Step 2: Introduce the slack or surplus variables, wherever necessary to convert the inequations

into equations and obtain the optimum solution of the given L.P.P. by using simplex algorithm.

Step 3: Test the integrality of the optimum solution

a) If the optimum solution contains all integer values, an optimum basic feasible integer solution

has been obtained.

b) If the optimum solution does not include all integer values then proceed onto next step.

Step 4: Examine the constraint equations corresponding to the current optimum solution. Let

these equations be represented by

Where n denotes the number of variables and m the number of equations.

Choose the largest fraction of bis ie to find {bi}i

Let it be [bk 1]

or write is as

f ko

Step 5: Express each of the negative fractions if any, in the k th row of the optimum simplex

table as the sum of a negative integer and a nonnegative fraction.

Step 6: Find the Gomorian constraint

and add the equation

to the current set of equation constraints.

Step 7: Starting with this new set of equation constraints, find the new optimum solution by dual

simplex algorithm. (So that Gsla (1) is the initial leaving basic variable).

Step 8: If this new optimum solution for the modified L.P.P. is an integer solution. It is also

feasible and optimum for the given I.P.P. otherwise return to step 4 and repeat the process until an

optimum feasible integer solution is obtained