DOMS304 INTRODUCTION TO OPERATIONS RESEARCH JULY-AUG 2025

Description

SESSION	JULY-AUG 2025
PROGRAM	MASTER OF BUSINESS ADMINISTRATION (MBA)
SEMESTER	III
COURSE CODE & NAME	DOMS304  INTRODUCTION TO OPERATIONS RESEARCH

 

 

 

Assignment Set – 1

 

 

Q1. Define Operations Research. Explain the nature and scope of Operations Research.   4+6

Ans 1.

Operations Research

Operations Research (OR) is a scientific and systematic approach to decision-making that uses mathematical models, analytical techniques, and quantitative tools to solve complex organizational problems. It focuses on identifying the optimal or most efficient course of action from a set of alternatives by applying logical reasoning, statistical analysis, and advanced computational methods. OR helps managers take objective, data-driven decisions in situations involving constraints, limited resources, and conflicting objectives.

At its core, Operations Research transforms real-world problems into structured models that can be analyzed and solved logically. Whether the issue involves minimizing cost,

 

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Q2. Solve the following linear programming problem using Revised Simplex Method:

    Maximize Z    = x1 + 9×2 + x3

    Subject to:          x1 + 2×2 + 3×3 ≤ 9

                             3×1 + 2×2 + 2×3 ≤ 15

                             2×1 + 3×2 + x3 ≤ 14

                             where x1, x2 ≥ 0 

 

 
Solution:
Problem is

Max Z = x1 + 9 x2 + x3

subject to

x1 + 2 x2 + 3 x3 ≤ 9 3 x1 + 2 x2 + 2 x3 ≤ 15 2 x1 + 3 x2 + x3 ≤ 14

and x1,x2,x3≥0;

Step-1 :

The problem is converted to canonical form by adding slack, surplus and artificial variables

 

 

 

Q3. Solve the following Linear Programming Problem using its Dual:

Minimize Z = 3×1 + 4×2

Subject to: 4×1 + x2 ≥ 30

                  -x1 – x2 ≤ -18 

                   x1 + 3×2 ≥ 28

                  and x1, x2 ≥ 0

             

 Ans 3.

Step 1: Write the Primal in Standard “Min” Form

Primal (Minimization):

Number of constraints = 3 → dual will have 3 variables

Number of variables in primal = 2 → dual will have 2 constraints.

So the Dual LPP is:

Subject to (columns of primal → rows of dual):

For : coefficients (4, 1, 1) and cost 3:

For : coefficients (1, 1, 3) and cost 4:

 

Step 2: Convert Dual to Standard Max Form

 

 

 

Assignment Set – 2

 

 

Q4. Five wagons are available at station 1, 2, 3, 4 and 5. These are required at five cities I, II, III, IV and V. The mileages between various stations and cities are given by the table below. How should the wagons be transported so as to minimize the total mileage covered?

 Ans 4.

Five wagons and five cities

Step 1: Mileage (Cost) Table

Five wagons are at stations 1–5 and required at cities I–V. The mileage (cost) matrix is:

Station \ City	I	II	III	IV	V
1	10	5	9	18	10
2	13	9	6	12	14
3	3	2	4	4	5
4	18	9	12	17	15
5	11	6	14	19	10

Let

Objective (minimize total mileage):

Subject to:

• Each station used once:
• Each city gets one wagon:

We now solve by the Hungarian Method.

Step 2: Row Reduction

 

 

 

Q5. Solve the following integer programming problem using Branch and Bound method:

Maximize Z = 7×1 + 6×2

Subject to: 2×1 + 3×2 ≤ 12

                  6×1 + 5×2 ≤ 30 

             and x1, x2 must be non-negative integers

             

 Ans 5.

Step 1: LP Relaxation

First we ignore the integer condition and solve the LP relaxation:

subject to

 

To find the LP optimum, consider the intersection of (1) and (2):

From (1):

From (2):

Solving simultaneously:

Multiply (1) by 3:

 

 

 

Q6. Write detail notes with application in management on the following:

1. i) Simulation Annealing Method
2. ii) Genetic Algorithm 5×2

Ans 6.

(i) Simulated Annealing Method

Simulated Annealing (SA) is a powerful probabilistic optimization technique inspired by the annealing process used in metallurgy, where metals are heated and gradually cooled to achieve a stable, low-energy molecular structure. In operations research, SA is used to find near-optimal solutions for complex optimization problems where traditional methods struggle due to multiple local optima. It explores the solution space by accepting both improving and, occasionally, non-improving solutions. This controlled randomness helps the algorithm escape local minima and continue the search for a global optimum. The method begins with an initial solution and a high “temperature.” As the temperature decreases based on a cooling