ME 324: Industrial Engineering and Operations Research

Last date for submission: 20-2-25 Assignment 5A Expected time to solve: 3 hours

Instructions: Section A students should submit the solutions to odd numbered questions and Section B students the
solutions to even numbered questions. You are free to take help from AI tools, internet and friends. However, you will
be responsible for any errors or copyright violation. It is suggested that you should answer in your words like a creative
author. At the beginning of the assignment, provide the following declaration: “I declare that I have taken the help
from following AI tools {Nil or names of tools}, web and book/paper sources {Nil or names of sources} and friends
{Nil or name of students with roll number}. Further, I have helped the following students in solving the assignment
{Names with roll number}. I understand that helping my colleagues is a noble activity and it will be appreciated by
the instructor. I spent {number of hours} in solving this assignment.” Write the answers with pen in good handwriting.

Q.1: A company wants to forecast its monthly sales for the next period based on past data. The
sales data (in units) for the last six months are as follows:

Month Sales (Units)

July 450
August 470
September 520
October 580
November 600
December 640
(a) Compute a 3-month moving average forecast for January.
(b) Use exponential smoothing with a smoothing constant α = 0.3 and an initial forecast for July
of 450 to forecast sales for January.
(c) Suppose the actual sales in January turn out to be 670 units. Compute the Mean Absolute
Error (MAE) based on the moving average and exponential smoothing forecasts.
Show all calculations clearly.
Solution:
(a) 3-Month Moving Average Forecast for January
The 3-month moving average is calculated as
Sales in October+Sales in November +Sales in December 580  600  640
Moving average=   606.67
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Hence, the 3-month moving average forecast for January is 606.67 units.

(b) Exponential Smoothing Forecast for January

The exponential smoothing formula is:

St   Dt  1    St 1 ,

where St 1 is the forecast for period t, Dt is the actual demand for the period t, St is the forecast
for period t+1 and α = 0.3 is the smoothing constant.
We are given that the initial forecast for July is 450. Now, we compute the forecast for each month.

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Forecast for August: SAug=0.3(450)+0.7(450)=450
Forecast for September: SSep=0.3(470)+0.7(450)=456
Forecast for October: SOct=0.3(520)+0.7(456)=475.2
Forecast for November: SNov=0.3(580)+0.7(475.2)=506.64
Forecast for December: SDec=0.3(600)+0.7(506.64)=534.648
Forecast for January: SJan=0.3(640)+0.7(534.648)=566.254
Hence, the exponential smoothing forecast for January is 566.254 units.

(c) Error Analysis (Mean Absolute Error - MAE)

Since we only have one forecasted value for January, the MAE values are:

 MAE (Moving Average) = (670606.67)= 63.33 units

 MAE (Exponential Smoothing) = (670566.254)=103.746 units

Error in Exponential Smoothing is more because there is increasing trend in demand. We could
have use higher value of  or used regression equation.
Q.3: A quality control engineer monitors the production of LED bulbs and inspects 100 bulbs
per day for defects. Over 10 days, the following number of defective bulbs were recorded:
Day 1 2 3 4 5 6 7 8 9 10
Defects 5 7 6 4 8 6 9 5 7 6
a. Calculate the centerline p for the p-chart.
b. Determine the upper control limit (UCL) and lower control limit (LCL).
c. Interpret whether the process is in control based on these limits.

Solution:

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Q.5: Three Ph.D. students X, Y and Z have to be assigned to three supervisors A, B and C.
Estimated number of papers each pair can produce in 5 years are given in the form of matrix.
How should I assign students to supervisors, such that department publishes the maximum
number of papers?

A B C
X 15 10 9
Y 9 15 10
Z 10 12 8

Solution:

This is a classic assignment problem, which can be solved using the Hungarian algorithm (or
other optimization techniques like integer programming). The goal is to maximize the number of
papers published by assigning students to supervisors in the best possible way.

Apply the Hungarian Algorithm (Maximization Case)

 Convert the problem into a minimization problem by subtracting each element from the
maximum element in the matrix.
 Solve the transformed problem using the Hungarian algorithm, which finds the optimal
assignment.
 Otherwise:
 The Hungarian Algorithm solves minimization problems, so we convert the given matrix
into a cost matrix by negating all values.
 Transformed matrix:

-15 -10 -9

-9 -15 -10

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 -10 -12 -8

 For each row, subtract the smallest value in that row from all elements in the row:

0 5 6

6 0 5

2 4 0

• For each column, subtract the smallest value in that column from all elements in the
column:

0 5 6

6 0 5

0 4 0

• Using the Hungarian Algorithm's zero-covering method, we determine the optimal

assignment:
• - Student X → Supervisor A
• - Student Y → Supervisor B
• - Student Z → Supervisor C
• Total maximum papers published: 38

Q.7. Explain exponential smoothing and moving average methods of forecasting.

Solution:
Moving average Method:
It is the simplest extrapolative method. Two steps are required to make a forecast for the
next period from past data.
1. Select the number of periods for which moving averages will be computed. This
number, N, is called an order of moving average.
2. Take the average demand for the most recent N periods. This average demand then
becomes the forecast for the next period.
Exponential Smoothing Method:
 In this method, the weight assigned to a previous period’s demand decreases exponentially
as that data gets older.
 The recent demand data receives a higher weight than older demand data.

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  Exponential smoothing methods are particularly attractive for production and operations
applications that involve forecasting for a large number of items.
 The forecasting horizon is relatively short (daily, weekly, or monthly demand needs to be
forecasted)
 This method is used when there is little “outside” information available about cause and
effect relationships between the demand for an item and independent factors that influence
it.
 The simplest exponential smoothing model is applicable when there is no trend or
seasonality component in the data.
 Only the horizontal component of demand is present. Because of randomness, the demand
fluctuates around an “average demand,” which is called as the “base.”
 If the base is constant from period to period, then all fluctuations in demand must be due
to randomness.
 In reality, fluctuations in demand are caused by both changes in the base and random noise.
 The key objective in exponential smoothing models is to estimate the base and use
that estimate for forecasting future demand.
 In the basic exponential smoothing model, the base for the current period, St, is estimated
as
New base = Previous base + (Actual current demand ‒ Previous base)
Q.9: The desired frequency of the power supply of equipment is 50 Hz. The permissible limit of
variation in frequency is 4%. Exceeding this limit, the equipment will be damaged causing a loss
of Rs. 20000/-. Considering the parabolic loss function of Taguchi, estimate the loss if the
frequency is 51 Hz.
Solution:

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Q.11: To make a product in mass production, the following tasks are needed:
Task number Task time (minutes)
1 37
2 5
3 3
4 35
5 9
6 17
7 14
8 25
9 30
What is the minimum possible cycle time? How many stations are needed to achieve it?
Solution:

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