MA IN LOGISTICS
MODULE: NUMERICAL METHODS
FORECASTING AND DECISION-MAKING IN LOGISTICS
MANAGEMENT: UTILIZING NUMERICAL TECHNIQUES
TABLE OF CONTENTS
Task 1: Introduction ........................................................................................................................ 3
Task 2: Problem with Manufacturing company production quantities ........................................... 5
Task 3: Derivative of the Inventory Function ................................................................................. 7
Task 4: Demand Forecasting Model ............................................................................................... 8
Task 5: Cost Function Analysis ...................................................................................................... 9
Task 6: Conclusion ....................................................................................................................... 13
Reference List ............................................................................................................................... 14
2
Task 1: Introduction
There are two different strategies for solving any problem concerning logistics management, the
analytical method and the numerical method. The two techniques are different, but problems are
solved analytically and numerically in logistics management. Analytical techniques involve the
use of problem-solving tools which are in the form of some mathematical formulas and equations
that provide final solutions (Hoseinzadeh et al. 2019). These methods are applied where the
problem is simple and can be formulated in terms of the formulas and the results are easily obtained
and understood. However, analytical methods are not very applicable in systems that have
nonlinearities, more than one variable, or are stochastic as it is very difficult, if not impossible, to
obtain solutions. On the other hand, numerical methods provide approximate solutions of the
mathematical models, by solving them iteratively with the help of computational techniques. These
methods are particularly useful in the management of logistics since the real problems are usually
too complex to be solved by analysis (Mohanty and Shankar, 2019). For example, in supply chain
network design, demand forecasting or inventory management the systems are often too complex
for analytical solutions. These situations can be solved by using numerical techniques such as
linear programming techniques, Monte Carlo simulation, and finite element technique and the
solution obtained is approximate but very useful in practical life and can be easily arrived at with
the help of computer programs available in the market (Edelkamp et al. 2016). However, the
analytical methods have limitations in the assumptions made and the approximations that are
required which can lead to less accurate or relevant solutions in real-life logistics problems. For
example, in the real world, logistics problems may be presented with the following constraints
such as the demand pattern, transport networks, and suppliers’ reliability among others that when
incorporated in the solutions give the problem a level of realism that cannot be tackled by analytical
approaches (Ghiani et al. 2022). This is why it is necessary to use numerical methods at all because
these factors cannot be set to a certain value and their variation can be described using iterations,
and simulations. Furthermore, the numerical methods are useful in handling problems of different
nature because of the flexibility of the methods. In the context of logistics management, this
flexibility is used in the feeding of real-time data and updating of the models with fresh data as
and when they are obtained. This flexibility is essential to the conditions of the logistics operations
because change is always bound to occur and decisions have to be made in the shortest time
possible to avoid disruption of the flow of goods. Thus, the integration of numerical methods
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contributes not only to the improvement of calculation accuracy but also to the actions’ flexibility
and the possibility of making decisions in the present conditions of logistics management.
The justification of the use of numerical methods in managing logistics can be explained by the
fact that the field of logistics is complex and unpredictable. The use of numerical techniques gives
the managers the most appropriate information, manner of operation, and how to handle change.
This capability is useful to have in a competitive context where it is necessary to modify logistics
systems to face new issues and opportunities. Hence, numerical methods are crucial in enhancing
the decision-making and forecasting on the management of logistics of organizations to compete.
4
Task 2: Problem with Manufacturing company production quantities
Firstly let us consider:
Variables
xA = Quantity of product A produced.
xB = Quantity of product B produced.
These are the decision variables that the quantities of products A and B to be produced. In logistics,
the right quantities to be produced for use in stock control, resource management and satisfaction
of customers have to be determined.
Resources
Now defining the resources:
The overall process of production mainly uses two different resources the se are law materials and
Labor
Now, let us consider
rA = Amount of raw material required per unit of product A.
rB = Amount of raw material required per unit of product B.
lA= Amount of labor required per unit of product A.
lB = Amount of labor required per unit of product B.
This is the total amount we have to sell and which has to be optimised. In this model it has been
assumed that both products are equally participating in the total output of the system.
Constraints
For defining the constraints for
R = Total amount of raw material available.
L = Total amount of labor available.
System Equation
Now formulating the System equations
Raw Material Constraint
rA . xA + rB . xB ≤ R
The above formula will ensure that the overall raw material that will be used by both products
will not exceed the Raw material that is available.
Labor Constraint
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lA . xA + lB . xB ≤ L
The above formula will ensure that the overall labor that will be used by both products will not
exceed the labor that is available.
Objective Functions
For maximizing the total output the objective function will be:
Maximize Z = xA + xB
Here, Z is the total production output for A and B products.
System of Equations
Using all information from the derived equations that will represent the production quantities of
both products is:
Maximize Z = xA + xB
Subject to:
rA . xA + rB . xB ≤ R
lA . xA + lB . xB ≤ L
xA ≥ 0, xB ≥ 0
6
Task 3: Derivative of the Inventory Function
Provided function:
I(t) = 2000 𝑒 −0.05𝑡
Here the I(t) represent the level of inventory at t time.
Now calculating the derivative:
To calculate the derivative of change in inventory over time I(t), it is required to calculate the
derivative
𝑑𝐼 (𝑡)
𝑑𝑡
𝑑𝐼 (𝑡)
𝑑𝑡
𝑑
= 𝑑𝑡 (2000 𝑒 −0.05𝑡 )
Using the chain rule,
𝑑𝐼 (𝑡)
𝑑𝑡
= 2000. (-0.05). 𝑒 −0.05𝑡
𝑑𝐼 (𝑡)
𝑑𝑡
= -100 𝑒 −0.05𝑡
Therefore, the derivative will be
𝑑𝐼 (𝑡)
𝑑𝑡
The derivative
𝑑𝐼 (𝑡)
𝑑𝑡
= -100 𝑒 −0.05𝑡
= -100 𝑒 −0.05𝑡 gives the rate of change with time of inventory. The negative
sign shows that as time goes up, the inventory level goes down this is normally so in most inventory
systems because of sales or usage of inventories.
At t = 0:
𝑑𝐼 (0)
𝑑𝑡
= -100 𝑒 0 = -100 units per unit time.
This means that at the beginning, the value of the inventory is declining by 100 units each time
period.
As t increases:
The term 𝑒 −0.05𝑡 reduces thus the rate at which the inventory is depleted is slower and slower over
time is the result of the use of the term. This is for instance where the demand or usage of the
inventory reduces over time or where the inventory in question is in a state where fewer items are
being removed from the stock.
Application in the Inventory Management:
The following are some of the reasons why knowledge of the rate of change of inventory is
important in inventory management:
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
Inventory Turnover Rates: The derivative is important in determining the rate of using,
selling or using up the inventories (Lee et al. 2018). The use of the upward slop depicts a
fast-moving product while the slop that is downwards depicts a slowing down product or
service. The information collected here can be used in reconsideration of reorder points so
that there is no repetitively large stock or small stock.

Demand Forecasting: Through this way, managers are in a position to predict the usage of
the inventory and make right assumptions of the necessity of the inventory in the future
(Wild, 2017). If the rate is slowing down, then it may be a signal that there is a need to
switch the buying or manufacturing strategies.

Cost Management: Inventory turnover is also another very critical area that should be
accorded a lot of attention since it is a direct determinant of holding cost and capital tied in
the inventories (Muchaendepi et al. 2019). The derivative provides information as to when
stock may become a problem or when the price of the products need to be lowered so that
consumers can be attracted.
Thus, the ‘inventory derivative’ is useful for the identification of the usage of inventories and help
the managers in making the right decisions in the area of inventory control, demand planning and
costs.
Task 4: Demand Forecasting Model
The selected data is:
Table 1: Historic data for forecasting
Month
Demand
January
120
February
130
March
125
April
140
May
145
June
150
July
160
August
170
September
165
8
October
175
November
180
December
190
(Source: Self-made in MS Word)
Using the linear interpolation method
The demand for December month was 190 and on the other hand the demand was 180 for
November, therefore the change demand is:
Δ = 190 − 180 = 10 Units
Therefore, linear interpolation model to predict January's demand will be:
𝐷𝑒𝑚𝑎𝑛𝑑𝐽𝑎𝑛 = 190 + Δ = 190 + 10 = 200 Units
To find the accuracy of the model
Error = Predicted – Actual = 200 – 195 = 5 Units
Percentage
5
Error Percentage = (195) ∗ 100 = 2.56%
There are many techniques of making forecasts of future demand, one of which is the linear
interpolation method being the simplest one. This method is however useful when the demand has
a relatively straight line and may not be very useful when the demand is erratic or has a seasonal
pattern. In the case, the interpolation model approximated the demand within a small range of error
(2.56%) this means that in those cases where demand changes are gradual and foreseeable, then
interpolation is actually quite helpful. However, if there are changes in the demand patterns such
as in the case of promotions, market conditions or changes in the economic situation, then the
simple models like moving averages, exponential smoothing or even the machine learning
techniques may be needed. Interpolation assists in demand forecasting since business needs to be
in a position to make good decisions on inventory that will enable it stock sufficient inventory to
meet the demand without having to stock excess inventory. This optimization helps the company
to minimize holding costs and at the same time the risk of stock out was also minimized thus
improving the supply chain.
Task 5: Cost Function Analysis
The provided cost function:
C (x) = √2
1
𝜋𝜎
𝑒
2
−
(𝑥− 𝜇)2
2𝜎2
+ 0.5𝑥 3 − 2𝑥 2 + 100
9
Here,
x is the quantity of a product
𝜇 is the mean = 10,
𝜎 = 2 which is the standard deviation.
Part A
To find the overall cost that is that is incurred over x when the range tends from 0 to 20. Using the
method of numeric integration of the provided cost function
Therefore, 𝐶𝑡𝑜𝑡𝑎𝑙 will be:
20
𝐶𝑡𝑜𝑡𝑎𝑙 = ∫ 𝐶(𝑥) 𝑑𝑥
0
Using the method of numeric Integration that is Trapezoidal Rule to solve the equation
(Chakraborty et al. 2021)
The interval [0, 20] is to be divided into n subintervals having the equal width
Δ (x) =
20−0
𝑛
Therefore, the Trapezoidal approximation integral is:
𝑛−1
Δx
𝐶𝑡𝑜𝑡𝑎𝑙 ≈
[𝐶(𝑥0 ) + 2 ∑ 𝐶(𝑥𝑖 ) + 𝐶(𝑥𝑛 )]
2
𝑖=1
Here 𝑥𝑖 = 0, Δx, 2Δx, … . , 20.
Now, we need to assume n = 4 so the Δx = 5 to calculate
C (0), C (5), C (10), C (15), and C (20)
Now, for C (0)
C (0) =
1
√2 𝜋(2)2
𝑒
−
(0− 10)2
2(2)2
+ 0.5(0)3 − 2(0)2 + 100
As 100/8 is 12.5 therefore 𝑒 −12.5 which is very close to 0
C (0) =
C (0) ≈
1
√8 𝜋
1
√8 𝜋
𝑒 −12.5 + 0 − 0 + 100
∗ 0 + 100 ≈ 100 units
Now for C (10)
C (10) =
1
√2 𝜋(2)2
𝑒
−
(10− 10)2
2(2)2
+ 0.5(10)3 − 2(10)2 + 100
10
C (10) =
As
1
√8 𝜋
1
√8 𝜋
+ 500 − 200 +100
≈ 0.141
C (10) = 0.141 + 500 − 200 +100 = 400.141 units
Now for C (20)
C (20) =
1
√2 𝜋(2)2
C (20) =
1
√8 𝜋
𝑒
−
(20− 10)2
2(2)2
+ 0.5(20)3 − 2(20)2 + 100
𝑒 −12.5 + 0.5 (8000) − 2(400) +100
C (20) ≈ 0 + 4000 – 800 + 100 = 3300 units
Part B
The information that is useful for managing inventories is the total cost of holding inventories
obtained through numerical integration of the mean value of the inventory levels over the range of
the inventory levels 0 to 20 units. This result embraces all holding costs, the shortage costs and
ordering costs throughout the whole range of inventory. Inventory control has it that there are
perceived to be varying costs of inventories (Ayoub et al. 2017). At lower levels of inventory, the
cost is mainly made up of the cost of stockouts, which is the loss of sales, the cost of handling back
orders and customer complaints. Fixed costs are directly related to the volumes of stock in that the
higher the volumes of stock the higher the fixed costs. These are cost of storage, insurance costs,
cost of depreciation due to obsolescence and the cost of funds which have been tied up through
inventory.
The total cost function can be integrated over a certain interval and the analysis of the result assists
to identify when the cost begins to increase. For example, when the integration result shows that
the total cost is rising steeply after a certain inventory level, this means that high inventory level
is unprofitable (Munir et al. 2020). This may be as a result of factors such as the cost of purchasing
more space to store other products, increased insurance expenses or even the probability of having
a certain stock that may take a long time to sell. On the other hand, if the cost line does not rise
and fall steeply with change in inventory it means that there is some freedom within that range of
inventory stock within the firm.
Besides, it assists the managers in evaluating the trade-offs as the total cost behaviour over the
range of inventory level is also determined. For instance, they may agree to pay a little more than
the optimal amount to holding costs in order to avoid stock outs if the integration result depict that
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the impact of the costs are insignificant. It is particularly important when defining the inventory
management policies, for instance when determining the safety stock quantities or when adjusting
the reorder points to the costs.
Part C
The general cost which is associated with changes in inventory levels is one of the key principles
of effective inventory control. The information derived from total cost function assists the
managers to make a rational decision on the number of inventories to order in a bid to minimize
the cost that is associated with holding inventories while satisfying the needs of the customers.
Firstly, as it will be seen from the total cost curve, the optimal level of inventory can be easily
identified. By identifying the point when the cost function rises, the managers are in a position to
set the inventory levels that do not put much cost while at the same time ensuring that the demand
is met (Dillon et al. 2017). This optimization eliminates situations where organizations hold large
stocks that tie down capital and raise holding cost at the same time as it eliminates situations where
there are no stocks to meet demand which leads to loss of sales and customer dissatisfaction.
Second, the total cost function is useful in determination of the reorder points Since the total cost
function is useful in determination of the reorder points, this paper seeks to estimate the total cost
function. The integration result can assist in explaining how cost varies with inventory level and
as such assist the managers to set the right reordering points that are near the right inventory levels
so as not to order excessively or insufficiently (Simchi‐Levi et al. 2018). This is relevant for
maintaining services in the long term and at the same time not to overpay significantly. For
instance, if the cost function indicates that the costs rise steeply with the inventory level it is better
to set a lower reorder point to avoid high inventory holding costs but at the same time ensure that
the organisation does not run out of stock. Thirdly, the total cost analysis can also be applied in
the setting of price strategies as well. If it is costly to hold inventory, the managers may use price
strategies that will encourage customers to stock more for instance offering a discount when
making more orders or having a discount sale of the products that are not selling fast in the shops.
Thus, pricing strategies can be aligned to the inventory costs in order to assist the firms to operate
more profitably and exercise greater inventory control. Moreover, knowledge of the total cost
function is important in capacity planning and in terms of budgeting (Taleizadeh et al. 2018). It is
much simpler for managers to identify the probability of costs that will be incurred in case of
having to purchase extra space, hiring more employees or purchasing improved systems of
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inventory control. These are long-run decisions and they assist the company in planning for the
future in such a way that it would not be in a one-point position to supply the market needs at very
high costs.
Overall it can be concluded that knowledge of the total cost function enables the management to
take the right decisions regarding the right inventory level and overall efficiency that help in
achieving the maximum profit. Through the implementation of all these findings in the inventory
management systems, firms are able to have a better balance of their stock and costs, hence,
enhancing the competitive advantage of firms in the market.
Task 6: Conclusion
Overall, it should be pointed out that all the numerical methods discussed in this paper starting
from linear programming ending up with the derivative analysis and demand forecasting are the
tools in the field of logistics management. They assist the managers to resolve the practical issues
which cannot be resolved using analytical tools and techniques. For example, in linear
programming, one is able to find the most suitable way of using resources for production and
consumption with least costs but maximum efficiency. Derivative analysis on inventory functions
is applied to explain the behaviors of inventory and make decisions on stock and turnover. In
contrast, the moving averages are a basic but effective method in demand forecasting that will
assist organisations to determine the future demand and order or build up inventory in advance.
These numerical techniques help the logistics managers in arriving at facts based decisions as to
their performance. With these tools, they can enhance the flow of work, reduce costs and, in
general, enhance the performance. In a field such as logistics, where there are so many factors, and
where these factors change so often, numerical methods bring the acuteness and the dynamism
that is required to meet the competition. As a result the use of such methods is crucial to facilitate
the manage of the logistics and supply chain processes in a way that is able to meet the ever
changing demands of the market.
13
Reference List
Journals
Hoseinzadeh, S., Heyns, P.S., Chamkha, A.J. and Shirkhani, A., 2019. Thermal analysis of porous
fins enclosure with the comparison of analytical and numerical methods. Journal of Thermal
Analysis and Calorimetry, 138, pp.727-735. https://link.springer.com/article/10.1007/s10973019-08203-x
Mohanty, M. and Shankar, R., 2019. A hierarchical analytical model for performance management
of
integrated
logistics.
Journal
of
Management
Analytics,
6(2),
pp.173-208.
https://www.academia.edu/download/98258646/23270012.2019.160832620230205-1q9c23b.pdf
Ghiani, G., Laporte, G. and Musmanno, R., 2022. Introduction to Logistics Systems Management:
With
Microsoft
Excel
and
Python
Examples.
John
Wiley
&
Sons.
https://books.google.com/books?hl=en&lr=&id=pzeVEAAAQBAJ&oi=fnd&pg=PR1&dq=diffe
rence+between+analytical+and+numerical+methods+in+logistics+management&ots=d_vug-4L9&sig=-H82s6gAKHUbBBNZcxBnz3Ups2k
Edelkamp, S., Gath, M., Greulich, C., Humann, M., Herzog, O. and Lawo, M., 2016. Monte-Carlo
tree search for logistics. In Commercial Transport: Proceedings of the 2nd Interdiciplinary
Conference on Production Logistics and Traffic 2015 (pp. 427-440). Springer International
Publishing. https://link.springer.com/chapter/10.1007/978-3-319-21266-1_28
Lee, C.K., Lv, Y., Ng, K.K., Ho, W. and Choy, K.L., 2018. Design and application of Internet of
things-based warehouse management system for smart logistics. International Journal of
Production
Research,
56(8),
pp.2753-2768.
https://ira.lib.polyu.edu.hk/bitstream/10397/73807/1/a0768-n06_1567.pdf
Wild,
T.,
2017.
Best
practice
in
inventory
management.
Routledge.
http://edl.emi.gov.et/jspui/bitstream/123456789/73/1/Best%20Practice%20in%20Inventory%20
Management.pdf
Muchaendepi, W., Mbohwa, C., Hamandishe, T. and Kanyepe, J., 2019. Inventory management
and performance of SMEs in the manufacturing sector of Harare. Procedia Manufacturing, 33,
pp.454-461.
https://www.sciencedirect.com/science/article/pii/S2351978919305335/pdf?md5=740b5d3fdab3
42a8b99db8cffa417cbe&pid=1-s2.0-S2351978919305335-main.pdf
14
Chakraborty, A., Mondal, S.P., Mahata, A. and Alam, S., 2021. Different linear and non-linear
form of trapezoidal neutrosophic numbers, de-neutrosophication techniques and its application in
time-cost optimization technique, sequencing problem. RAIRO-Operations Research, 55, pp.S97S118. https://www.rairo-ro.org/articles/ro/pdf/2021/01/ro190161.pdf
Ayoub, H.F., Abdallah, A.B. and Suifan, T.S., 2017. The effect of supply chain integration on
technical innovation in Jordan: the mediating role of knowledge management. Benchmarking: An
International
Journal,
24(3),
pp.594-616.
https://www.emerald.com/insight/content/doi/10.1108/BIJ-06-2016-0088/full/html
Munir, M., Jajja, M.S.S., Chatha, K.A. and Farooq, S., 2020. Supply chain risk management and
operational performance: The enabling role of supply chain integration. International Journal of
Production
Economics,
227,
p.107667.
https://www.academia.edu/download/103036751/j.ijpe.2020.10766720230605-1-yjb912.pdf
Dillon, M., Oliveira, F. and Abbasi, B., 2017. A two-stage stochastic programming model for
inventory management in the blood supply chain. International Journal of Production Economics,
187,
pp.27-41.
https://optimise.amsi.org.au/wp-content/uploads/sites/52/2017/07/dillon-ellie-
uom-opt17-talk.pdf
Simchi‐Levi, D., Wang, H. and Wei, Y., 2018. Increasing supply chain robustness through process
flexibility and inventory. Production and Operations Management, 27(8), pp.1476-1491.
https://dspace.mit.edu/bitstream/handle/1721.1/140771/10.1111poms.12887.pdf?sequence=1&isAllowed=y
Taleizadeh, A.A., Soleymanfar, V.R. and Govindan, K., 2018. Sustainable economic production
quantity models for inventory systems with shortage. Journal of cleaner production, 174, pp.10111020.
https://portal.findresearcher.sdu.dk/files/135803718/The_link_between_response_time_and_pref
erence_variance_and_processing_heterogeneity_in_stated_choice_experiments.pdf
15