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Inventory optimization using calculus

(Combination of economics and mathematics)

IA

Maths AA SL

Rationale

The inventory management internship that I completed during my winter break provided me

with valuable practical experience and a deeper understanding of the day-to-day operations of

inventory management. The internship was a great opportunity to apply the theoretical concepts

I learned in my academic coursework to real-world situations.

During the internship, I had the opportunity to work closely with the inventory management

team, assisting with tasks such as data entry, order fulfilment, and supply chain management.

Through these tasks, I gained a better understanding of how inventory management affects the

overall performance of a business, including its profitability and customer satisfaction.

In addition to the technical skills I gained during the internship, I also developed important soft

skills such as communication, teamwork, and problem-solving. Working alongside

experienced professionals provided me with valuable mentorship and guidance, helping me to

hone my skills and develop a stronger sense of professional identity.

Overall, the inventory management internship was an enriching experience that helped me to

develop my skills and knowledge in a practical, hands-on setting. I believe that the experience

will serve me well as I move forward in my academic and professional career, and I am grateful

for the opportunity to have participated in the internship.

Introduction

In order to maximize operations and make wise choices regarding stock levels, purchases, and

sales, firms rely heavily on mathematical ideas in inventory management. The significance of

mathematical ideas in inventory management and how they might be used to raise effectiveness

and profitability are covered in this IA.

Monitoring the movement of items from suppliers to customers is the process of inventory

management. It entails keeping an eye on inventory levels, figuring out the ideal order size,

and making sure there is adequate stock to fulfil demand from customers. Businesses must

comprehend and evaluate data linked to stock levels, sales, and demand using mathematical

ideas in order to make informed decisions regarding inventory (Sethi, 2021).

Additionally, businesses may benefit from using mathematical models like forecasting theory

such as Markov chains to better understand how inventory systems behave and forecast future

demand. An inventory system's performance may be evaluated using any of these ideas, and

decisions on how to enhance it can then be made. Economic order quantity is a further

important mathematical idea in inventory management (EOQ). The optimal order quantity

undergone numerous revisions since then. Demand, ordering, and holding expenses are all

considered fixed in the economic order number calculation.

By using data-driven approaches and mathematical techniques, businesses can minimize the

risks of overstocking and understocking, optimizing their operations and increasing

profitability. I thought that was incredibly fascinating and tried to go further into that. As a

part of my IA, I chose to analyse various inventory models (De-La-Cruz-Márquez et al., 2021).

For that I studied two models of inventory management, basic model to optimize the inventory

quantity and minimize the cost associated with it. I used applications of calculus to determine

the optimum value, and using matrix property concavity and convexity of a functions are

verified.

Aim

My research aim is to answer following research questions.

RQ1: How much to order or to produce for each inventory cycle?

RQ2: When to order or produce a new quantity?

Mathematical Exploration of the IA

Finding the maximum or minimum value of a certain quantity is desirable in many practical

situations. These applications may be found in engineering, business, and economics.

Numerous problems may be resolved utilizing differential calculus techniques.

Calculus plays a crucial role in the EOQ model, as it provides the mathematical framework for

finding the minimum of the total cost function. Calculus is used to find the minimum point of

the total cost function by taking its first derivative with respect to the order quantity, setting it

to zero, and solving for the order quantity that minimizes the function. This process involves

using calculus concepts such as the chain rule and the product rule.

Focus of this study is mainly on economic aspects so maxima and minima may be used, in

particular, to:

• maximize the positive values, such as profit, efficiency, production of a firm,
• minimize the negative values, such as costs, efforts,
• be used in the study of inventory control theories and economic order quantity model.

Finding Tangents using First Derivative

Determining the absolute and relative maxima and minima of distinct functions constitutes a

significant class of problems that may be solved using differential calculus. The idea that a

function's derivative is equal to the slope of its tangent is the foundation for the theory used to

determine a function's maximum and lowest values. A function is considered to be raising when

its values rise as the value of the independent variable rises and the lines tangent to the

function's graph have a positive slope. In contrast, the tangent lines have a negative slope and

the function is said to be lowering when the values of the function decline as the independent

variable's values increase. The tangent line is horizontal (has slope 0), and the derivative is zero

precisely where the function switches from rising slope to dropping slope either from dropping

to rising (De-La-Cruz-Márquez et al., 2021).

Finding the turning points when the function output has a maximum value or a minimum value

is assisted by the first derivative test. Regarding the initial derivative test. If a function f(x) is

continuous at a critical point c in the interval itself and is applied to an open interval I.

• If f ′(x) shifts from a +ve to a - ve sign as x is rising in till c, i.e., if f ′(x) > 0 at every position

satisfactorily close and is at the left of c and less than 0 at every point similarly close to and to

the right of c, then c is a point of local maximum.

• If f ′(x) shifts from a - ve to a +ve sign as x is rising in till c, i.e., if f ′(x) < 0 at every

satisfactorily close and is at the left of c and f ′(x) > 0 at every point at the right hand side of c,

then c is a point is called as point of local minima.

• If the value of f ′(x) does not noticeably change as x is rising in till c, then c is neither a

location of local maxima nor minima. Such a place is really referred to as an inflection point.

Figure 2: First order derivative test using tangent property

Inventory Model:

Basic Model

One of the common inventory control analyses is described in this section. In order to determine

how much to order, it demonstrates how we might balance the various stock costs. The strategy

is to create a model of a perfect inventory structure and determine the fixed order quantity

(FOQ) to reduce the overall expenses. The economic order quantity is the name given to this

optimal order size (EOQ). The EOQ design is a crucial inventory control analysis, and it is also

arguably one of the most crucial operations management outcomes. Other sources give Wilson

(1934), who conducted the investigation on his own and sold the findings, credit for the

computation. The first mention of the work was made by Harris (1915). I begin with a

straightforward model and make the following assumptions: (1) that the demand is precisely

acknowledged, constant, and continuous over time; (2) that all prices are identified exactly so

and do not vary; (3) that no shortages are permitted; and (4) that the lead time has to be nil, i.e.,

that a delivery is made immediately after the order is placed.

Finally, we always place orders of precisely this amount since I am searching for the fixed

ordering quantity that will reduce expenses. The final pattern is displayed in fig. 3.

Figure 3: Represents a constant demand over a period of time [2]

Figure 4: Stock level with fixed order size [2]

Nomenclature:

Types of Costs in Inventory Management [4,5]

Unit cost

(UC)

Price agreed by the suppliers for a single unit of the good or the total

cost incurred by the organization to purchase a single unit.

Reorder cost

(RC)

Costs associated with placing a recurring order for the goods, including

possible charges for phone calls, receiving, delivering swiftly, doing

quality inspections, etc.

Holding cost

(HC)

is the cost of holding one item in stock for a specific amount of time.

The standard unit of measurement for stock costs is a year.

Shortage cost

(SC)

is the cost of not having enough inventory to meet demand and

experiencing a deficit.

Types of Variable we have considered in this Inventory Management [4,5]

Order quantity

(Q)

It is the predetermined order quantity we always use. This investigation's

objective is to determine the optimal value for this order quantity.

Cycle time

(T)

This is the space in between two replenishments that happen back to

back. Larger orders have longer cycle times, therefore this depends on

the quantity of the order.

Demand

(D)

which determines the number of items to be delivered from stock within

a particular time limit (for example, ten units a week). In this instance,

we'll assume that there is a consistent and ongoing need

Only the order quantity, which we are free to set at any number, is directly within our control.

This fixes the cycle length when we set the order quantity. Since we have no influence over the

other variables, we make the assumption that they are all fixed. In light of these other constants,

our goal is to determine the best values for Q and, subsequently, T.

Gives

𝑇𝐶 = (𝑈𝐶 × 𝑄) + (𝑅𝐶) + E𝐻𝐶 ×

× 𝑇F

Eq. 6

Step: 2 The second step calculates the cost per unit of time by dividing it by the cycle length,

T.

𝑇𝐶 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑡𝑖𝑚𝑒 = 𝑈𝐶 ×

+ 𝐻𝐶 ×

Eq. 7

But according to Eq. 1 𝑄 = 𝐷 × 𝑇,

And rewriting the Eq. 7 by eliminating the time (T) terms

𝑅𝐶 × 𝐷

Eq. 8

Step 3: The overall minimum cost per unit hour is determined in the third step of our analysis.

𝑇𝐶 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑡𝑖𝑚𝑒 has three components,

The unit cost component (𝑈𝐶 × 𝐷) is independent of order quantity and is ‘fixed.

The other two components H

!" ×%

&

'"&

(

I vary with order quantity and cost.

In this : The 𝐻𝐶 component increases linearly through Q

Though the 𝑅𝐶 component drops as Q rises.

To find the optimal order quantity I just have to focus on the components having variable effect

with quantity.

For this I differentiate the Eq. 8 with respect to Q and compare the first derivative with:

𝑅𝐶 × 𝐷

E𝑈𝐶𝐷 +

𝑅𝐶 × 𝐷

F = 0

E 0 −

𝑅𝐶 × 𝐷

(

F = 0

(

(

L

2 × 𝑅𝐶 × 𝐷

Eq. 9

To analyze the data, I first had to collect and organize it into a manageable format. This

involved creating spreadsheets and databases to store the data, and then inputting the data into

these tools. I then calculated the average product consumption over a period, as well as the

frequency and quantity of orders, here in table 1 data for selected 11 products are shown.

Table 1 Consumption data of various products

1 2 3 5 7 8 9 10 11 12 13 14 15 16 17 19

No. Descriptions Item Quantity April May June July August Sept Oct Nov Dec Jan Feb Total yearly

consumption

Type

1 BUY 140.

280 280 280 280 280 280 280 280 280 280 280 3360

2 BUY 5.

10 10 10 10 10 10 10 10 10 10 10 120

3 BUY 90.

180 180 180 180 180 180 180 180 180 180 180 2160

4 BUY 7.

14 14 14 14 14 14 14 14 14 14 14 168

5 BUY 225.

450 450 450 450 450 450 450 450 450 450 450 5400

6 BUY 2.

4 4 4 4 4 4 4 4 4 4 4 48

7 BUY 35.

70 70 70 70 70 70 70 70 70 70 70 840

8 BUY 530.

1060 1060 1060 1060 1060 1060 1060 1060 1060 1060 1060 12720

9 BUY 6.

12 12 12 12 12 12 12 12 12 12 12 144

10 BUY 885.

1770 1770 1770 1770 1770 1770 1770 1770 1770 1770 1770 21240

11 BUY 120.

240 240 240 240 240 240 240 240 240 240 240 2880

By analysing the data, I was able to identify patterns in product consumption and ordering, and

gain insights into the effectiveness of the inventory management system. I also identified areas

where the system could be improved, such as reducing lead times for orders and optimizing

stock levels.

Here, I am presenting how I did the calculation for one product called journal bearing. The

yearly consumption of this item is 2500 sets per year. It costs them 10 AED to store one set for

Figure 6: Graph of total cost function

Calculus method

The minimum ordering cost value of 𝐹(𝑥) in the region of [1, 2500] is obtained by doing first

order derivative as

)

(

)

𝑒xists for all values of 𝑥 in 1 ≤ 𝑥 ≤ 2500 ;

[

]

We need those critical answer of x is obtained by equating first order derivative with zero,

)

)

(

(

Since I am focusing on only one critical value in [1, 2500] in the positive region. I used the

second derivative to check whether it produces a maximum or a minimum value,

))

))

(𝑥) is +ve for all x in [1, 2500]

So I took the minimum value as 𝑥 = 100. Thus, to minimize inventory costs, the store should

order or 25/year. Substituting the answer of 𝑥 = 100 in total cost will be 23500 AED. The

sample calculation for one product is shown and the same calculations were repeated for all 15

types of components are calculated in table 2.

Table 2 Calculation of EOQ

20 21 22 23 24

Growth Ordering cost

Storage

cost

Installation

Cost EOQ

10% AED AED AED Quantity

3696 13 20 8 770

132 17 19 5 28

2376 8 16 14 495

184.8 14 24 17 39

5940 21 17 0 1238

52.8 05 13 0 11

924 32 40 3 193

13992 27 15 8 2915

158.4 17 28 6 33

23364 24 21 7 4868

3168 11 18 8 660

This EOQ calculation can help a company to,

Reduce inventory costs: EOQ calculation helps companies identify the ideal order quantity that

minimizes the costs of carrying inventory. This can help a company reduce the costs associated

with holding excess inventory, such as storage, insurance, and maintenance costs.

Improve order frequency: EOQ calculation can help a company optimize the ordering

frequency, ensuring that the company always has sufficient stock to meet customer demand

while avoiding overstocking. This can help improve operational efficiency and customer

satisfaction.

Optimize inventory management: EOQ calculation provides a scientific approach to inventory

management, allowing companies to better manage their inventory levels, reduce stock outs,

Moreover, it may be helpful to consider areas where I could have improved or expanded upon

my work. For example, were there any limitations to your data or assumptions that could have

impacted your results? Were there other inventory management models or strategies that as

could have explored?

Overall, this IA enhance my deeper knowledge of optimization on business practices on

inventory management and the EOQ model likely provided valuable insights into a critical

aspect of business operations. As I continue to develop my skills and knowledge in this area, I

may find new and innovative ways to optimize inventory and maximize efficiency.

Conclusion

In conclusion, mathematical ideas play a crucial part in inventory management by assisting

companies in making wise choices regarding stock levels, purchases, and sales. Businesses

may streamline processes, cut expenses, and boost profitability by utilizing probability, linear

programming, Markov chains, queueing theory, EOQ, and statistical analysis. Businesses may

make sure they are using their resources as efficiently as possible and are in a good position to

satisfy their customers' requirements by implementing these mathematical ideas into their

inventory management procedures.

Economic ordering quantity (EOQ) is a term used to describe the lot size that reduces overall

inventory expenses.

The method described above makes three assumptions in order to calculate the economic

ordering quantity, and this model can only predict the right values under these conditions.

• To begin with, the product's demand remains constant throughout the year. This may be

reasonable for things with a fixed demand, like televisions, but it is implausible for items that

are seasonal, like apparel or skis.

• Second, throughout the year, there is no variation in the time it takes to receive an order after

placing one.

• Lastly, the prices involved are consistent across the board, including those for shipping,

storage, and other fees. In an era of inflation, this presumption might not be valid, but variations

in these prices can be accommodated by speculating on what they might be and utilizing

average costs.

However, the model presented above is helpful since it enables us to utilize mathematics to

analyze an apparently challenging problem.

Future work for Extension of the study

The application of mathematical optimization techniques such as linear programming, dynamic

programming, and stochastic programming will be increasingly important in developing

optimal inventory management strategies. These techniques can be used to identify the best

trade-offs between different inventory costs, such as ordering costs, holding costs, and stock-

out costs, and to identify the optimal order quantity and reorder point that minimize the total

inventory costs.

In inventory management is the use of simulation modelling to test different scenarios and

evaluate the impact of different inventory management strategies. Mathematical models such

as queuing theory and Markov chains can be used to develop realistic simulations of supply

chain systems, which can be used to evaluate different inventory management strategies and

identify areas for improvement.