Description

Operations Research
April 2024 Examination
Q1. A retail company operates several distribution centers (D1, D2, D3, and D4) and
serves multiple retail stores (S1, S2, and S3). The shipping costs (in Rs.) per unit from
each distribution center to each retail store are presented in the following table
D1 D2 D3 D4 Supply
S1 19 30 50 10 7
S2 70 30 40 60 9
S3 40 8 70 20 18
Demand 5 8 7 14
Find the initial basic feasible solution using Vogel’s Approximation Method (VAM) for
the given transportation problem. Post that, implement the stepping-stone method to
find the optimal solution for the transportation problem. Calculate the total shipping
cost for the optimal solution. (10 marks)
Ans 1.
To find the initial basic feasible solution using Vogel’s Approximation Method (VAM), we
start by calculating the penalty for each row and column. The penalty for each row is the
difference between the two lowest costs in that row, and the penalty for each column is the
difference between the two lowest costs in that column. We will use these penalties to
determine which cells to allocate first.
Step 1: Calculate Row and Column Penalties
Row Penalties: For S1: Penalty = 30 – 19 = 11
For S2: Penalty = 70 – 30 = 40 For S3: Penalty = 70 – 20 = 50
Column Penalties: For
D1: Penalty = 19 – 10 = 9
For D2: Penalty = 8 – 30 = -22 For
It is only half solved
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Q2. A manufacturing company has several plants (P1, P2, and P3) and several
warehouses
(WH1, WH2, WH3, and WH4) for distribution. The shipping costs (in Rs.) per unit
from each plant to each warehouse are presented in the following table:
Source WH1 WH2 WH3 WH4 Supply
P1 19 30 50 12 7
P2 70 30 40 60 10
P3 40 10 60 20 18
Requirement 5 8 7 15
Find an initial feasible solution using the Northwest Corner Method and Least
Cost Method, and also determine the optimal solution using Modified Distribution
(MODI) method. (10 marks)
(Note- For each method, show the step-by-step calculations, allocations, and the total
transportation cost for the final optimal solution)
Ans 2.
To solve this transportation problem, we will use three methods as you requested: the
Northwest Corner Method, the Least Cost Method, and the Modified Distribution (MODI)
Method. We’ll calculate the total transportation cost for each method. Let’s begin:
1. Northwest Corner Method
The Northwest Corner Method is a way to find an initial feasible solution for a transportation
problem. We start from the top-left (northwest) corner of the cost matrix and allocate as much
Q3. A small project consisting of eight activities has the following characteristics: Time-
Estimates (in weeks)
Activity Preceding
activity
Most optimistic
time (a)
Most likely
time (m)
Most pessimistic
A None 2 4 t1i2me (b)
B None 10 12 26
C A 8 9 10
D A 10 15 20
E A 7 7.5 11
F B,C 9 9 9
G D 3 3.5 7
H E,F,G 5 5 5
PART a) Prepare the activity schedule for the project and determine the critical path.
(5 marks)
Ans 3a.
Introduction
In project management, creating an activity schedule and determining the critical path are
essential steps in planning and executing a project efficiently. The given project, consisting of
eight activities with varying time estimates, requires careful analysis to establish the sequence
and duration of each activity. By applying the Critical Path Method (CPM), we can identify
PART b) Suppose a 30-week deadline is imposed, what is the probability that the
project will be finished within the time limit? Also, if the project manager wants to be
99% sure that the project is completed on the scheduled date, how many weeks before
that date should he start the project work? (5 marks)
Ans 3b.
To determine the probability that the project will be finished within the time limit, we can use
the completion probability of the critical path activities. The completion probability of an
activity is the probability that the activity will be completed on time, given its estimated
duration and the time limit.
The completion probability of an activity is calculated as follows:
Completion Probability = 1 – (1 / Durations in weeks)