Description

SESSION JUL – AUG 2024
PROGRAM MASTER OF BUSINESS ADMINISTRATION
(MBA)
SEMESTER 3
COURSE CODE & NAME DOMS304 APPLICATIONS OF OPERATIONS
RESEARCH

Assignment Set – 1

1. A factory manufactures two products A and B. To manufacture one unit of A, 10
machine hours and 15 labour hours are required. To manufacture product B, 20
machine hours and 15 labour hours are required. In a month, 400 machine hours and
300 labour hours are available. Profit per unit for A is Rs. 75 and for B is Rs. 50.
Formulate as LPP.
Ans 1.
Formulating the Linear Programming Problem (LPP)
In this scenario, a factory manufactures two products, A and B, using limited resources:
machine hours and labor hours. The aim is to determine the optimal production quantities of
these products to maximize profit while staying within the resource constraints. This problem
can be formulated as a Linear Programming Problem (LPP) as follows:
Decision Variables
To represent the quantities of the two products, we define:
 x1: Number of units of product A to be produced.
 x2: Number of units of product B to be produced.
These variables must satisfy the constraints imposed by the resource availability and cannot
Its Half solved only
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2. Find solution using Simplex method
MAX Z = 3×1 + 5×2 + 4×3
subject to
2×1 + 3×2 <= 8
2×2 + 5×3 <= 10
3×1 + 2×2 + 4×3 <= 15
and x1,x2,x3 >= 0
Ans 2.
Problem is
Max Z= 3 x1+5 x2+4 x3
subject to
2 x1+3 x2 ≤ 8
2 x2+5 x3 ≤ 10
3 x1+2 x2+4 x3 ≤ 15
and x1,x2,x3≥0;
The problem is converted to canonical form by adding slack, surplus and artificial variables
as appropiate
1. As the constraint-1 is of type ‘≤’ we should add slack variable S1
2. As the constraint-2 is of type ‘≤’ we should add slack variable S2
3. As the constraint-3 is of type ‘≤’ we should add slack variable S3
After introducing slack variables
Max Z= 3 x1+5 x2+4 x3+0S1+0S2+0S3
subject to
2 x1+3 x2 + S1 =8
2 x2+5 x3 + S2 =10
3 x1+2 x2+4 x3 + S3=15
3. Solve the following LPP graphically
Max Z = 4x + 5y
Subject to
x + y ≤ 20
3x + 4y ≤ 72
x, y ≥ 0
Ans 3.
Problem is
MAX Z= 4 x1+5 x2
subject to
x1+ x2 ≤ 20
3 x1+4 x2 ≤ 72
and x1,x2≥0;
Hint to draw constraints
1. To draw constraint x1+x2≤20→(1)
Treat it as x1+x2=20
When x1=0 then x2=?
⇒(0)+x2=20
⇒x2=20
When x2=0 then x1=?
⇒x1+(0)=20
⇒x1=20
x1 0 20
x2 20 0
Put x1=0,×2=0 (origin) in x1+x2≤20, then 0+0≤20, which is true,
Assignment Set – 2
4. Obtain an optimum solution to the following transportation problem
Factory Warehouse Capacity
W1 W2 W3 W4
F1 19 30 50 10 7
F2 70 30 40 60 9
F3 40 8 70 20 18
Requirements 5 8 7 14
Ans 4.
Step-by-Step Solution to the Transportation Problem
Problem Data:
Factory
Warehouse
W1
Warehouse
W2
Warehouse
W3
Warehouse
W4 Capacity
F1 19 30 50 10 7
F2 70 30 40 60 9
Factory
Warehouse
W1
Warehouse
W2
Warehouse
W3
Warehouse
W4 Capacity
F3 40 8 70 20 18
Warehouse Requirement
W1 5
W2 8
W3 7
W4 14
Step 1: Define the Decision Variables
Let xij represent the number of goods transported from factory Fi to warehouse Wj.
Step 2: Objective Function
Minimize the total cost:
Z = 19×11 + 30×12 + 50×13 + 10×14 + 70×21 + 30×22 + 40×23 + 60×24 + 40×31 + 8×32
+ 70×33 + 20×34
5. Consider the problem of assigning five jobs to five persons. The assignment costs are
given as follows. Determine the optimum assignment schedule.
Job
Person 1 2 3 4 5
A 8 4 2 6 1
B 0 9 5 5 4
C 3 8 9 2 6
D 4 3 1 0 3
E 9 5 8 9 5
Ans 5.
Problem: Assignment of Jobs to Persons
The task involves assigning five jobs to five persons such that the total assignment cost is
minimized. The cost matrix is given as:
Job 1 Job 2 Job 3 Job 4 Job 5
A 8 4 2 6 1
B 0 9 5 5 4
C 3 8 9 2 6
D 4 3 1 0 3
E 9 5 8 9 5
We solve this using the Hungarian Method, which can be implemented algorithmically
6. Discuss the applications of Integer programming.
Ans 6.
Applications of Integer Programming
Integer Programming (IP) is a specialized field within optimization that focuses on problems
requiring decision variables to take integer values. It is widely applied in various industries
and sectors due to its ability to address real-world problems where solutions must be discrete,
such as scheduling, allocation, and resource optimization. Below are key applications of
Integer Programming, discussed in detail:
1. Supply Chain Management
Integer Programming is extensively used in supply chain optimization to address problems