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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