Quantitative Methods Linear Programming

This document discusses the concept of linear programming in quantitative methods and provides an optimal loan plan based on certain assumptions. It also includes a cash flow summary and sensitivity analysis.

Running head: QUANTITATIVE METHODS LINEAR PROGRAMMING
Quantitative Methods Linear Programming
Name of the Student:
Name of the University:
Author’s Note:

1QUANTITATIVE METHODS LINEAR PROGRAMMING
Table of Contents
Answer 1. A.....................................................................................................................................2
Answer 1. B.1..................................................................................................................................3
Answer 1. B.2..................................................................................................................................3
Answer 1. C.....................................................................................................................................4
Answer 2..........................................................................................................................................5
Answer 3. A.....................................................................................................................................6
Answer 3. B.....................................................................................................................................8
Reference:........................................................................................................................................9

2QUANTITATIVE METHODS LINEAR PROGRAMMING
Answer 1. A
Let’s assume
L1 = Annual loan amount
S1= monthly loan for January
S2= monthly loan for February
S3= monthly loan for March
S4= monthly loan for April
S5= monthly loan for May
S6= monthly loan for June
S7= monthly loan for July
S8= monthly loan for August
S9= monthly loan for September
S10= monthly loan for October
S11= monthly loan for November
S12= monthly loan for December
C1= Net Operating Cash flow for January
C2 Net Operating Cash flow for February

3QUANTITATIVE METHODS LINEAR PROGRAMMING
C3= Net Operating Cash flow for March
C4= Net Operating Cash flow for April
C5= Net Operating Cash flow for May
C6= Net Operating Cash flow for June
C7= Net Operating Cash flow for July
C8= Net Operating Cash flow for August
C8= Net Operating Cash flow for September
C10= Net Operating Cash flow for October
C11= Net Operating Cash flow for November
C12= Net Operating Cash flow for December
The linear equation for this case study on the basis of the above assumptions is as
follows:
C12-(L1*1.12)-(S12*1.025) = 8000
Constraints:
L1, S1, S2, S3, S4, S5, S6, S7, S8, S9, S10, S11 and S12 >= 0
Table 1: Optimal Loan Plan
Optimal Loan Plan:
Annual Loan Monthly Loan Total Loan
January $ 4,047.63 $ 0.00 $ 4,047.63
February $ 0.00 $ 0.00 $ 0.00

4QUANTITATIVE METHODS LINEAR PROGRAMMING
March $ 0.00 $ 952.37 $ 952.37
April $ 0.00 $ 976.18 $ 976.18
May $ 0.00 $ 0.59 $ 0.59
June $ 0.00 $ 0.00 $ 0.00
July $ 0.00 $ 0.00 $ 0.00
August $ 0.00 $ 0.00 $ 0.00
September $ 0.00 $ 2,000.60 $ 2,000.60
October $ 0.00 $ 2,050.62 $ 2,050.62
November $ 0.00 $ 0.05 $ 0.05
December $ 0.00 $ 0.00 $ 0.00
TOTAL $ 4,047.63 $ 5,980.41 $ 10,028.03
Working
CASH FLOW SUMMARY:
Particulars January February March April May June July August September October November December Total
Cash Flow from Operating Activity:
Sales Revenue $ 4,000.00 $ 3,000.00 $ 3,000.00 $ 3,000.00 $ 5,000.00 $ 9,000.00 $ 3,000.00 $ 2,000.00 $ 1,000.00 $ 2,000.00 $ 6,000.00 $ 10,000.00 $ 51,000.00
Bills ($ 6,000.00) ($ 5,000.00) ($ 4,000.00) ($ 3,000.00) ($ 4,000.00) ($ 1,000.00) ($ 6,000.00) ($ 6,000.00) ($ 4,000.00) ($ 2,000.00) ($ 1,000.00) ($ 1,000.00) ($ 43,000.00)
Net Operating Cash Flow ($ 2,000.00) ($ 2,000.00) ($ 1,000.00) $ 0.00 $ 1,000.00 $ 8,000.00 ($ 3,000.00) ($ 4,000.00) ($ 3,000.00) $ 0.00 $ 5,000.00 $ 9,000.00 $ 8,000.00
Cash Flow from Investing Activity:
Annual Loan $ 4,047.63 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 4,047.63
Monthly Loan $ 0.00 $ 0.00 $ 952.37 $ 976.18 $ 0.59 $ 0.00 $ 0.00 $ 0.00 $ 2,000.60 $ 2,050.62 $ 0.05 $ 0.00 $ 5,980.41
Interest on Loan $ 0.00 $ 0.00 ($ 23.81) ($ 24.40) ($ 0.01) $ 0.00 $ 0.00 $ 0.00 ($ 50.02) ($ 51.27) ($ 485.72) ($ 635.23)
Repayment of Principal Loan $ 0.00 $ 0.00 ($ 952.37) ($ 976.18) ($ 0.59) $ 0.00 $ 0.00 $ 0.00 ($ 2,000.60) ($ 2,050.62) ($ 4,047.68) ($ 10,028.03)
Net Cash Flow from Investing Activity $ 4,047.63 $ 0.00 $ 952.37 ($ 0.00) ($ 1,000.00) ($ 0.60) $ 0.00 $ 0.00 $ 2,000.60 ($ 0.00) ($ 2,101.83) ($ 4,533.39) ($ 635.23)
Net Cash Flow $ 2,047.63 ($ 2,000.00) ($ 47.63) ($ 0.00) ($ 0.00) $ 7,999.40 ($ 3,000.00) ($ 4,000.00) ($ 999.40) ($ 0.00) $ 2,898.17 $ 4,466.61 $ 7,364.77
Add: Opening Balance $ 0.00 $ 2,047.63 $ 47.63 $ 0.00 $ 0.00 ($ 0.00) $ 7,999.40 $ 4,999.40 $ 999.40 $ 0.00 $ 0.00 $ 2,898.17 $ 0.00
Closing Balance $ 2,047.63 $ 47.63 $ 0.00 $ 0.00 ($ 0.00) $ 7,999.40 $ 4,999.40 $ 999.40 $ 0.00 $ 0.00 $ 2,898.17 $ 7,364.77 $ 7,364.77
Answer 1. B.1
Table 2: Optimal Loan Plan When the Interest Rate is 10.5%
Sensitivity Summary 1:
Annual Loan Monthly Loan Total Loan Annual Loan Monthly Loan Total Loan Variance
January $ 4,047.63 $ 0.00 $ 4,047.63 $ 4,047.63 $ 0.00 $ 4,047.63 $ 0.00
February $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00
March $ 0.00 $ 952.37 $ 952.37 $ 0.00 $ 952.37 $ 952.37 $ 0.00
April $ 0.00 $ 976.18 $ 976.18 $ 0.00 $ 976.18 $ 976.18 $ 0.00
May $ 0.00 $ 0.59 $ 0.59 $ 0.00 $ 0.59 $ 0.59 $ 0.00
June $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00
July $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00
August $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00
September $ 0.00 $ 2,000.60 $ 2,000.60 $ 0.00 $ 2,000.60 $ 2,000.60 $ 0.00
October $ 0.00 $ 2,050.62 $ 2,050.62 $ 0.00 $ 2,050.62 $ 2,050.62 $ 0.00
November $ 0.00 $ 0.05 $ 0.05 $ 0.00 $ 0.05 $ 0.05 $ 0.00
December $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00
TOTAL $ 4,047.63 $ 5,980.41 $ 10,028.03 $ 4,047.63 $ 5,980.41 $ 10,028.03 $ 0.00
Optional Case 1 Base Case

5QUANTITATIVE METHODS LINEAR PROGRAMMING
Working
CASH FLOW SUMMARY:
Particulars January February March April May June July August September October November December Total
Cash Flow from Operating Activity:
Sales Revenue $ 4,000.00 $ 3,000.00 $ 3,000.00 $ 3,000.00 $ 5,000.00 $ 9,000.00 $ 3,000.00 $ 2,000.00 $ 1,000.00 $ 2,000.00 $ 6,000.00 $ 10,000.00 $ 51,000.00
Bills ($ 6,000.00) ($ 5,000.00) ($ 4,000.00) ($ 3,000.00) ($ 4,000.00) ($ 1,000.00) ($ 6,000.00) ($ 6,000.00) ($ 4,000.00) ($ 2,000.00) ($ 1,000.00) ($ 1,000.00) ($ 43,000.00)
Net Operating Cash Flow ($ 2,000.00) ($ 2,000.00) ($ 1,000.00) $ 0.00 $ 1,000.00 $ 8,000.00 ($ 3,000.00) ($ 4,000.00) ($ 3,000.00) $ 0.00 $ 5,000.00 $ 9,000.00 $ 8,000.00
Cash Flow from Investing Activity:
Annual Loan $ 4,047.63 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 4,047.63
Monthly Loan $ 0.00 $ 0.00 $ 952.37 $ 976.18 $ 0.59 $ 0.00 $ 0.00 $ 0.00 $ 2,000.60 $ 2,050.62 $ 0.05 $ 0.00 $ 5,980.41
Interest on Loan $ 0.00 $ 0.00 ($ 23.81) ($ 24.40) ($ 0.01) $ 0.00 $ 0.00 $ 0.00 ($ 50.02) ($ 51.27) ($ 425.00) ($ 574.51)
Repayment of Principal Loan $ 0.00 $ 0.00 ($ 952.37) ($ 976.18) ($ 0.59) $ 0.00 $ 0.00 $ 0.00 ($ 2,000.60) ($ 2,050.62) ($ 4,047.68) ($ 10,028.03)
Net Cash Flow from Investing Activity $ 4,047.63 $ 0.00 $ 952.37 ($ 0.00) ($ 1,000.00) ($ 0.60) $ 0.00 $ 0.00 $ 2,000.60 ($ 0.00) ($ 2,101.83) ($ 4,472.68) ($ 574.51)
Net Cash Flow $ 2,047.63 ($ 2,000.00) ($ 47.63) ($ 0.00) ($ 0.00) $ 7,999.40 ($ 3,000.00) ($ 4,000.00) ($ 999.40) ($ 0.00) $ 2,898.17 $ 4,527.32 $ 7,425.49
Add: Opening Balance $ 0.00 $ 2,047.63 $ 47.63 $ 0.00 $ 0.00 ($ 0.00) $ 7,999.40 $ 4,999.40 $ 999.40 $ 0.00 $ 0.00 $ 2,898.17 $ 0.00
Closing Balance $ 2,047.63 $ 47.63 $ 0.00 $ 0.00 ($ 0.00) $ 7,999.40 $ 4,999.40 $ 999.40 $ 0.00 $ 0.00 $ 2,898.17 $ 7,425.49 $ 7,425.49
From the above loan structure it is observed that the initial loan structure for
question 1.A has not changed due to the fall in the interest rate on annual loan. Hence it can be
said that there is no impact of the change in rate of interest on the optimal loan plan.
Answer 1. B.2
Table 3: Impact on Total amount of Loan Plan
Sensitivity Summary 2: Base Case Optional Case 2
Total Intrest Total Intrest Variance
January $ 0.00 $ 0.00 $ 0.00
February $ 0.00 $ 0.00 $ 0.00
March $ 0.00 $ 0.00 $ 0.00
April ($ 23.81) ($ 23.80) ($ 0.01)
May ($ 24.40) ($ 24.39) ($ 0.01)
June ($ 0.01) $ 0.00 ($ 0.01)
July $ 0.00 $ 0.00 $ 0.00
August $ 0.00 $ 0.00 $ 0.00
September $ 0.00 $ 0.00 $ 0.00
October ($ 50.02) ($ 50.00) ($ 0.02)
November ($ 51.27) ($ 88.75) $ 37.48
December ($ 485.72) ($ 485.78) $ 0.07
TOTAL ($ 635.23) ($ 672.72) $ 37.49

6QUANTITATIVE METHODS LINEAR PROGRAMMING
Working
CASH FLOW SUMMARY:
Particulars January February March April May June July August September October November December Total
Cash Flow from Operating Activity:
Sales Revenue $ 4,000.00 $ 3,000.00 $ 3,000.00 $ 3,000.00 $ 5,000.00 $ 9,000.00 $ 3,000.00 $ 2,000.00 $ 1,000.00 $ 2,000.00 $ 6,000.00 $ 10,000.00 $ 51,000.00
Bills ($ 6,000.00) ($ 5,000.00) ($ 4,000.00) ($ 3,000.00) ($ 4,000.00) ($ 1,000.00) ($ 6,000.00) ($ 6,000.00) ($ 4,000.00) ($ 3,500.00) ($ 1,000.00) ($ 1,000.00) ($ 44,500.00)
Net Operating Cash Flow ($ 2,000.00) ($ 2,000.00) ($ 1,000.00) $ 0.00 $ 1,000.00 $ 8,000.00 ($ 3,000.00) ($ 4,000.00) ($ 3,000.00) ($ 1,500.00) $ 5,000.00 $ 9,000.00 $ 6,500.00
Cash Flow from Investing Activity:
Annual Loan $ 4,048.19 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 4,048.19
Monthly Loan $ 0.00 $ 0.00 $ 951.81 $ 975.61 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 2,000.00 $ 3,550.00 $ 0.00 $ 0.00 $ 7,477.42
Interest on Loan $ 0.00 $ 0.00 ($ 23.80) ($ 24.39) $ 0.00 $ 0.00 $ 0.00 $ 0.00 ($ 50.00) ($ 88.75) ($ 485.78) ($ 672.72)
Repayment of Principal Loan $ 0.00 $ 0.00 ($ 951.81) ($ 975.61) $ 0.00 $ 0.00 $ 0.00 $ 0.00 ($ 2,000.00) ($ 3,550.00) ($ 4,048.19) ($ 11,525.61)
Net Cash Flow from Investing Activity $ 4,048.19 $ 0.00 $ 951.81 ($ 0.00) ($ 1,000.00) $ 0.00 $ 0.00 $ 0.00 $ 2,000.00 $ 1,500.00 ($ 3,638.75) ($ 4,533.97) ($ 672.72)
Net Cash Flow $ 2,048.19 ($ 2,000.00) ($ 48.19) ($ 0.00) ($ 0.00) $ 8,000.00 ($ 3,000.00) ($ 4,000.00) ($ 1,000.00) ($ 0.00) $ 1,361.25 $ 4,466.03 $ 5,827.28
Add: Opening Balance $ 0.00 $ 2,048.19 $ 48.19 $ 0.00 $ 0.00 ($ 0.00) $ 8,000.00 $ 5,000.00 $ 1,000.00 $ 0.00 ($ 0.00) $ 1,361.25 $ 0.00
Closing Balance $ 2,048.19 $ 48.19 $ 0.00 $ 0.00 ($ 0.00) $ 8,000.00 $ 5,000.00 $ 1,000.00 $ 0.00 ($ 0.00) $ 1,361.25 $ 5,827.28 $ 5,827.28
From the above loan structure it is observed that the initial loan structure for
question 1.A has changed due to the change in bill payment of October. Due to increase in the
bill payment in October from 2000 to 3500, the interest will fall by $37.49 and difference of
interest is majorly seen in the month of November.
Answer 1. B.3
Table 4: Loan Plan for the Revised Forecast Sales of September
Sensitivity Summary 3:
Annual Loan Monthly Loan Total Loan Annual Loan Monthly Loan Total Loan Variance
January $ 4,048.19 $ 0.00 $ 4,048.19 $ 4,047.63 $ 0.00 $ 4,047.63 $ 0.56
February $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00
March $ 0.00 $ 951.81 $ 951.81 $ 0.00 $ 952.37 $ 952.37 ($ 0.56)
April $ 0.00 $ 975.61 $ 975.61 $ 0.00 $ 976.18 $ 976.18 ($ 0.57)
May $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.59 $ 0.59 ($ 0.59)
June $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00
July $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00
August $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00
September $ 0.00 $ 500.00 $ 500.00 $ 0.00 $ 2,000.60 $ 2,000.60 ($ 1,500.60)
October $ 0.00 $ 512.50 $ 512.50 $ 0.00 $ 2,050.62 $ 2,050.62 ($ 1,538.12)
November $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.05 $ 0.05 ($ 0.05)
December $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00
TOTAL $ 4,048.19 $ 2,939.92 $ 6,988.11 $ 4,047.63 $ 5,980.41 $ 10,028.03 ($ 3,039.93)
Optional Case 3 Base Case
Working

7QUANTITATIVE METHODS LINEAR PROGRAMMING
CASH FLOW SUMMARY:
Particulars January February March April May June July August September October November December Total
Cash Flow from Operating Activity:
Sales Revenue $ 4,000.00 $ 3,000.00 $ 3,000.00 $ 3,000.00 $ 5,000.00 $ 9,000.00 $ 3,000.00 $ 2,000.00 $ 2,500.00 $ 2,000.00 $ 6,000.00 $ 10,000.00 $ 52,500.00
Bills ($ 6,000.00) ($ 5,000.00) ($ 4,000.00) ($ 3,000.00) ($ 4,000.00) ($ 1,000.00) ($ 6,000.00) ($ 6,000.00) ($ 4,000.00) ($ 2,000.00) ($ 1,000.00) ($ 1,000.00) ($ 43,000.00)
Net Operating Cash Flow ($ 2,000.00) ($ 2,000.00) ($ 1,000.00) $ 0.00 $ 1,000.00 $ 8,000.00 ($ 3,000.00) ($ 4,000.00) ($ 1,500.00) $ 0.00 $ 5,000.00 $ 9,000.00 $ 9,500.00
Cash Flow from Investing Activity:
Annual Loan $ 4,048.19 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 4,048.19
Monthly Loan $ 0.00 $ 0.00 $ 951.81 $ 975.61 $ 0.00 $ 0.00 $ 0.00 $ 0.00 $ 500.00 $ 512.50 $ 0.00 $ 0.00 $ 2,939.92
Interest on Loan $ 0.00 $ 0.00 ($ 23.80) ($ 24.39) $ 0.00 $ 0.00 $ 0.00 $ 0.00 ($ 12.50) ($ 12.81) ($ 485.78) ($ 559.28)
Repayment of Principal Loan $ 0.00 $ 0.00 ($ 951.81) ($ 975.61) $ 0.00 $ 0.00 $ 0.00 $ 0.00 ($ 500.00) ($ 512.50) ($ 4,048.19) ($ 6,988.11)
Net Cash Flow from Investing Activity $ 4,048.19 $ 0.00 $ 951.81 ($ 0.00) ($ 1,000.00) $ 0.00 $ 0.00 $ 0.00 $ 500.00 ($ 0.00) ($ 525.31) ($ 4,533.97) ($ 559.28)
Net Cash Flow $ 2,048.19 ($ 2,000.00) ($ 48.19) ($ 0.00) ($ 0.00) $ 8,000.00 ($ 3,000.00) ($ 4,000.00) ($ 1,000.00) ($ 0.00) $ 4,474.69 $ 4,466.03 $ 8,940.72
Add: Opening Balance $ 0.00 $ 2,048.19 $ 48.19 $ 0.00 $ 0.00 ($ 0.00) $ 8,000.00 $ 5,000.00 $ 1,000.00 ($ 0.00) ($ 0.00) $ 4,474.69 $ 0.00
Closing Balance $ 2,048.19 $ 48.19 $ 0.00 $ 0.00 ($ 0.00) $ 8,000.00 $ 5,000.00 $ 1,000.00 ($ 0.00) ($ 0.00) $ 4,474.69 $ 8,940.72 $ 8,940.72
From the above loan structure it is observed that the initial loan structure for
question 1.A has changed due to the change in the forecast of sales in September. Due to
increase in the forecast of sales in September from 1000 to 2500, the amount of loan will fall by
$3039.39 and the difference in loan is mainly seen in the month of September and October.
Answer 2.A
Linear programming is a widely used mathematical modelling technique that is operated
to derive the optimum result or solution subject to various constraint that are represented in a
linear relationship. This technique is extensively used for guiding quantitative conclusions in
industrial business planning, engineering. This technique is precisely used in social and physical
sciences. It is mostly used in the computer modelling in order to find the best results in allocating
known number of resources for example money, energy and manpower (Barnes et al. 2016).
Answer 2.B
Linear programming is basically used as a mathematical technique to determine and plan
for best results. This is mostly used to plan optimum expenditures and returns in such a way that
reduces the costs for the armed forces and caused the opposite for the enemy. Linear
programming is a major area in mathematics that is well known as optimization technique as it is
used to calculate the most optimum solution to a given problem (Tuohy et al. 2015). A very

8QUANTITATIVE METHODS LINEAR PROGRAMMING
basic example of linear optimization usage is in logistics or the “method of moving things
around efficiently. For an example, suppose there are 1500 baskets of equal size of 0.5 cubic
meter each, 6 pickup trucks that are able to carry 90 baskets, 50 baskets and 10 baskets
respectively. There are a number of routes and 36 hours to deliver all the baskets. Linear
programming helps to calculate the optimum number of pickup truck loading and the best route
that minimizes the time spending of transportation to transport the basket from one place to
another place. The linear programming gives an equation and provides the solution that the
transportation will be complete with minimum transportation cost and fastest time possible. The
basic units of linear programming are decision variable, objective function, constraint and data.
 Decision variables are the variables that needs to be derived.
 Objective function shows the influence of the decision variables on the values which
needs to be determined.
 Constraints shows the limitation of resources that are used by decision variables.
 Data measures the relation of objective function with constraints.
There are some assumption that needs to be fulfilled for effective results. The assumptions are:
 The objective function needs to be linear and the decisions variables are linear with
constraints.
 Values of decision variables are acceptable when they are non-integer.
 Parameters are known or given and must be constant.
 Negative values are not allowed to be taken.

9QUANTITATIVE METHODS LINEAR PROGRAMMING
Answer 3. A
There are a number of methods to forecast such as moving average, exponential
smoothing, regression and multiplicative decomposition. Here, these methods are used to
forecast and MAD, MSE and MAPE indicators are calculated to determine the efficient method
to forecast monthly lettings in 2019. The below table presents the MAD, MSE and MAPE
indicators for the forecasting methods.
Table 5: Indicators for the Forecasting Methods.
Moving
Average
Exponential
Smoothing Regression
Multiplicative
Forecast
MAD 51067.75567 45476.77839 56921.92157 45887.79083
MSE 67641.75559 59799.55167 71798.67428 59496.55378
MAPE 5.03% 4.42% 6.09% 5.15%
The above table shows that the exponential smoothing method is best in this case as the
value of mean absolute deviation is less and the mean absolute percentage error is very low.
Table 6: Standard Error of the Absolute Error for the Corresponding Year.
Year Mean SD SE
2007 45224.41 33636.86 10141.9
2008 39044.59 30519.68 8810.272
2009 44217.93 32166.01 9285.527
2010 48839.23 39214.74 11320.32
2011 39053.87 35617.08 10281.76
2012 33458.27 27550.95 7953.273
2013 46723.33 32593.5 9408.934
2014 44277.86 35903.01 10364.3
2015 69733.89 39544.59 11415.54
2016 51316.51 64876.22 18728.15

10QUANTITATIVE METHODS LINEAR PROGRAMMING
2017 45320.62 46618.66 13457.65
2018 31502.83 43783.46 17874.52
The above table shows the value of standard error of absolute error for the year 2016,
2017 and 2018 is high. This indicates that these three years are irrelevant and the year 2007,
2008, 2009, 2010, 2011, 2012, 2013, 2014 and 2015 are the relevant years for the forecasting of
the monthly paid lettings in 2019.
Answer 3. B
To forecast the monthly paid lettings for the year 2019, exponential smoothing is used.
To determine the efficient damping value, MAD, MSE and MAPE is calculated. The below table
shows the lowest MAD and MAPE for α=0.5.
Table 7: MAD, MSE and MAPE Corresponding to Different Damping Values
α=0.4 α=0.5 α=0.6
MAD 46464.69 45476.78 45574.50
MSE 60929.82 59799.55 59473.90
MAPE 4.52% 4.42% 4.44%
This implies that the exponential smoothing with the dumping value 0.5 gives the best
result for this specific case. This method gives the immediate next month’s forecast value which
is constant for all the next months. That means the forecast value of July, 2018 is 1487106 and
the forecast value for all the upcoming months is same. Therefore, the forecast value of each
months in 2019 is 1487106.

11QUANTITATIVE METHODS LINEAR PROGRAMMING
The limitation of the method is that the forecast value for next month is constant for the
whole year.

12QUANTITATIVE METHODS LINEAR PROGRAMMING
Reference:
Barnes, G., Leka, K.D., Schrijver, C.J., Colak, T., Qahwaji, R., Ashamari, O.W., Yuan, Y.,
Zhang, J., McAteer, R.T.J., Bloomfield, D.S. and Higgins, P.A., 2016. A Comparison of flare
forecasting methods. I. Results from the “all-clear” workshop. The Astrophysical
Journal, 829(2), p.89.
Tuohy, A., Zack, J., Haupt, S.E., Sharp, J., Ahlstrom, M., Dise, S., Grimit, E., Mohrlen, C.,
Lange, M., Casado, M.G. and Black, J., 2015. Solar forecasting: methods, challenges, and
performance. IEEE Power and Energy Magazine, 13(6), pp.50-59.