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(Book ID: B1631)Assignment Set - 1 (60 Marks)Note: Assignment Set -1 must be written within 6-8 pages. Answer all questions.

Q1. a. What do you mean by linear programming problem? Explain the steps involved in linear programming problem formulation? Ans 1 a)

Linear programming is a mathematical method for determining a way to achieve the best outcome in a

given mathematical model for some list of requirements represented as linear relationships. Linear

programming is a specific case of mathematical programming .

More formally, linear programming is a technique for the optimization of a linear objective function,

subject to linear equality and linear inequality constraints. Its feasible region is a convex polyhedron,

which is a set defined as the intersection of finitely many half spaces, each of which is defined by a

linear inequality. Its objective function is a real-valued affine function defined on this polyhedron. A

programming algorithm finds a point in the polyhedron where this function has the smallest value if

such a point exists.

A few examples of problems in which LP has been successfully applied are:

Developments of a production schedule that will satisfy future demands for a firm’s product and

at the same time minimize total production and inventory costs.

Establishment of an investment portfolio from a variety of stocks or bonds that will maximize a

company’s return on investment.

Allocation of a limited advertising budget among radio, TV, and newspaper spots in order to

maximize advertising effectiveness.

Determination of a distribution system that will minimize total shipping cost from several

warehouses to various market locations.

Selection of the product mix in a factory to make best use of machine and man hours available

while maximizing the firm’s profit.

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/in equations 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

b. A paper mill produces two grades of paper viz., X and Y. Because of raw material restrictions, it cannot produce more than 400 tons of grade X paperand 300 tons of grade Y paper in a week. There are 160 production hours in aweek. It requires 0.20 and 0.40 hours to produce a ton of grade X and Y papers.The mill earns a profit of Rs. 200 and Rs. 500 per ton of grade X and Y paperrespectively. Formulate this as a Linear Programming Problem.

Ans 1 b)

Max Z = 200x1 + 500x2

Subject To

X1 ≤ 400 (Quantity constraint)

X2 ≤ 300 (Quantity constraint)

0.20X1 + 0.40X2 ≤ 160 (Time constraint)

X1 ≥ 0 , X2 ≥ 0 (Non negativity constraint)

Q2. a. Discuss the methodology of Operations Research.

Ans

Operations Research MethodologyThe basic dominant characteristic feature of operations research is that it employs mathematical

representations or models to analyze problems. This distinct approach represents an adaptation of the

scientific methodology used by the physical sciences. The scientific method translates a given problem

into a mathematical representation which is solved and retransformed into the original context.

OR methodology consists of five steps. They are - defining the problem, constructing the model,

solving the model, validating the model, and implementing the result.

1) Definition

The first and the most important step in the OR approach of problem solving is to define the problem.

One needs to ensure that the problem is identified properly because this problem statement will

indicate the following three major aspects:

Description of the goal or the objective of the study

Identification of the decision alternative to the system

Recognition of the limitations, restrictions, and requirements of the system

2) Construction

Based on the problem definition, you need to identify and select the most appropriate model to

represent the system. While selecting a model, you need to ensure that the model specifies

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

variables. A model gives a perspective picture of the whole problem and helps in tackling it in a well-

organized manner. Therefore, if the resulting model fits into one of the common mathematical models,

you can obtain a convenient solution by using mathematical techniques. If the mathematical

relationships of the model are too complex to allow analytic solutions, a simulation model may be more

appropriate. Hence, appropriate models can be constructed.

3) Solution

After deciding on an appropriate model, you need to develop a solution for the model and interpret 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 an optimum solution. An optimum solution is one which

maximizes or minimizes the performance of any measure in a model subject to the conditions and

constraints imposed on the model.

4) Validation

A model is a good representation of a system. However, the optimal solution must work towards

improving the system’s performance. You can test the validity of a model by comparing its performance

with some past data available from the actual system. If under similar conditions of inputs, your model

can reproduce the past performance of the system, then you can be sure that your model is valid.

However, you will still have no assurance that future performance will continue to duplicate the past

behavior. Secondly, since the model is based on careful examination of past data, the comparison

should always reveal favorable results. In some instances, this problem may be overcome by using

data from trial runs of the system. One must note that such validation methods are not appropriate for

non-existent systems because data will not be available for comparison.

5) Implementation

You need to apply the optimal solution obtained from the model to the system and note the

improvement in the performance of the system. You need to validate this performance check under

changing conditions. To do so, you need to translate these results into detailed operating instructions

issued in an understandable form to the individuals who will administer and operate the recommended

system. The interaction between the operations research team and the operating personnel reaches its

peak in this phase.

b. Explain in brief the phases of Operations Research.Ans

Phases of Operations Research

The scientific method in OR study generally involves three phases.

Judgment phase

This phase includes the following activities:

Determination of the operations

Establishment of objectives and values related to the operations

Determination of suitable measures of effectiveness

Formulation of problems relative to the objectives

Research phase

This phase utilizes the following methodologies:

Operation and data collection for a better understanding of the problems

Formulation of hypothesis and model

Observation and experimentation to test the hypothesis on the basis of additional data

Analysis of the available information and verification of the hypothesis using pre-established

measure of effectiveness

Prediction of various results and consideration of alternative methods

Action phase

This phase involves making recommendations for the decision process. The recommendations can be

made by those who identify and present the problem or by anyone who influences the operation in

which the problem has occurred.

Q3. Solve the following Linear Programming Problem using Simple method.Maximize Z= 3x1 + 2X2Subject to the constraints:X1+ X2 ≤ 4X1+ X2 ≤ 2X1, X2 ≥ 0

Ans

X1 + X2 + S1= 4X1 – X2 + S2= 2-3X1 – 2X2 + Z= 0

  1 1 1 0 0 4→ 1 -1 0 1 0 2

  -3 -2 0 0 1 0  ↑    First Pivot Table

→ 0 2 1 -1 0 2  1 -1 0 1 0 2  0 -5 0 3 1 6    ↑    Second pivot table

0 1 1/2 - 1/2 0 11 0 1/2 1/2 0 30 0 2 1/2 3 1 11

Third Pivot Table

X1 = 3, X2 = 1, Max Z = 11

Q4. Explain the procedure of MODI method of finding solution through optimalitytest.

Ans 4

After evaluating an initial basic feasible solution to a transportation problem, the next question is how to

get the optimum solution. The basic techniques are illustrated as follows:

1. Determine the net evaluations for the non–basic variables (empty cells)

2. Determine the entering variable

3. Determine the leaving variable

4. You repeat steps 1 to 3 to till all allocations are over.

5. For allocating all forms of equations ui+ vj = cj, set one of the dual variable ui / vj to zero and solve

for others.

6. Use this value to find cij = cij - ui - vj. If all ij ≥ 0, then it is the optimal solution.

7. If any ij ≤ 0 select the most negative cell and form loop. Starting point of the loop is positive and

alternative corners of the loop are negative and positive. Examine the quantities allocated at negative

places. Select the minimum, add it to the positive places and subtract from the negative places.

8. Form a new table and repeat steps 5 to 7 till ij ≥ 0

Q5. a. Explain the steps in Hungarian method.

Ans 5 a

Hungarian Method Algorithm

Hungarian method algorithm is based on the concept of opportunity cost and is more efficient in solving

assignment problems. The following steps are adopted to solve an AP using the Hungarian method

algorithm.

Step 1: Prepare row ruled matrix by selecting the minimum values for each row and subtract it from the

other elements of the row.

Step 2: Prepare column-reduced matrix by subtracting minimum value of the column from the other

values of that column.

Step 3: Assign zero row-wise if there is only one zero in the row and cross (X) or cancel other zeros in

that column.

Step 4: Assign column wise if there is only one zero in that column and cross other zeros in that row.

Step 5: Repeat steps 3 and 4 till all zeros are either assigned or crossed. If the number of assignments

is equal to number of rows present, you have arrived at an optimal solution, if not, proceed to step 6.

Step 6: Mark the unassigned rows. Look for crossed zero in that row. Mark the column containing the

crossed zero. Look for assigned zero in that column. Mark the row containing assigned zero. Repeat

this process till all the makings are done.

Step 7: Draw a straight line through unmarked rows and marked column. The number of straight line

drawn will be equal to the number of assignments made.

Step 8: Examine the uncovered elements. Select the minimum.

Subtract it from the uncovered elements.

Add it at the point of intersection of lines.

Leave the rest as is.

Prepare a new table.

Step 9: Repeat steps 3 to 7 till optimum assignment is obtained.

Step 10: Repeat steps 5 to 7 till number of allocations = number of rows.

The assignment algorithm applies the concept of opportunity costs. The cost of any kind of action or

decision consists of the opportunities that are sacrificed in taking that action

Q5 B. Solve the following assignment problem.Machine Operators1 2 3 5A 60 50 45 45B 40 45 55 35C 55 70 60 50D 45 45 40 45

Ans b

Machine Operator  1 2 3 4A 60 50 45 45B 40 45 50 35C 55 70 60 50D 45 45 40 45

Machine Operator  1 2 3 4A 15 5 0 0B 5 10 20 0C 5 20 10 0D 5 5 0 5

Row Reduced Matrix

Machine Operator  1 2 3 4A 10 0 0 0B 0 5 20 0C 0 15 10 0D 0 0 0 5

Column Reduced Matrix

Since the no. of assignments is 4.A to 3 45B to 4 35C to 1 55D to 2 45

180Total Cost

Q6. a. Explain the steps involved in Vogel’s approximation method (VAM) of

solving Transportation Problem.

Ans 6 a)Vogel’s approximation methodThe Vogel’s approximation method (VAM) takes into account not only the least cost cij, but also the cost that just exceeds cij. The steps of the method are given as follows:

Step 1 - For each row of the transportation table, identify the smallest and the next to smallest costs. Determine the difference between them for each row. Display them alongside the transportation table by enclosing them in parenthesis against the respective rows. Similarly, compute the differences for each column.

Step 2 - Identify the row or column with the largest difference among all the rows and columns. If a tie occurs, use any arbitrary tie breaking choice. Let the greatest difference correspond to the ith row and let Cij be the smallest cost in the ith row. Allocate the maximum feasible amount xij = min (ai, bj) in the (i, j)th cell and cross off the ith row or the jth column in the usual manner.

Step 3 - Recomputed the column and row differences for the reduced transportation table and go to step 2. Repeat the procedure until all the rim requirements are satisfied.

b. Solve the following transportation problem using Vogel’s approximationmethod.Factories Distribution Centres SupplyC1 C2 C3 C4F1 3 2 7 6 50F2 7 5 2 3 60F3 2 5 4 5 25Requirements 60 40 20 15

Ans

Factories Distribution centres Supply Row Differencec1 c2 c3 c4      

F1 3(10) 2(40) 7 6 50 1 3 x

F2 7(25) 5 2(20) 3(15) 60 1 1 1

F3 2(25) 5 4 5 25 2 2 2

Requirements 60 40 20 15 135

Column Difference

1 3 2 2  

1 x 2 2  

5 x 2 2  

Total Cost = 2x40 + 3x10 + 2x25 + 7x25 + 2x 20 + 3x15 =420