# Central Limit Theorem for Process Improvement with Python

Estimate the workload for returns management assuming a normal distribution of the number of items per carton received from your stores.

Estimate the workload for returns management assuming a normal distribution of the number of items per carton received from your stores.

*Article originally published on Medium. *

** Returns management**, often referred to as

**is the management of returned items from retail locations in**

**reverse logistics,****.**

**your distribution centre**After the reception, products are sorted, organized, and inspected for quality. If they are in good condition, these products can be restocked in the warehouse and added to the inventory count waiting to be reordered.

In this article, we will see how the ** Central Limit Theorem** can help us to estimate the

**for the**

**workload****using a**

**process of returns management****based on the**

**normal distribution****of historical records.**

**mean and the standard deviation**

**SUMMARY**

**I. Scenario**

**Problem Statement**As the Inbound Manager of a multinational clothing retail company you are in charge of workforce planning for returns management.

**Question**Can you estimate the probability to have less than 30 items per carton that you will receive every week?

**II. Central Limit Theorem**1. Definition

2. Application

3. Probability to get <30 items per carton?

4. 95% probability to have less than k items per case?

**III. Conclusion**## I. Scenario

Problem Statement

You are the ** Inbound Manager of a multinational clothing retail company** known for its fast-fashion clothing for men, women, teenagers, and children.

A major problem for you is the ** lack of visibility of your workload for the returns process**. Indeed, because of system limitations, you do not get advance shipping notice (ASN) before receiving returns from your stores.

- You receive the cartons by pallets that you unload from the truck

2. You open the box and inspect the returned items

For each item (shirt, dress â€¦) your operators need to perform:

- Quality check to ensure that the product can be restocked
- Relabelling
- Re-packing

** You know the productivity per item** and you would like to estimate the

**based on the**

**workload in hours****.**

**number of cases you will receive in the week**Based on the historical data of the last 24 months, you have:

- An average of
**23 items per carton** - A standard deviation of
**7 items**

Your team is usually sized to handle a maximum of ** 30 items per case**. You

**.**

**need to hire temporary workers to meet your daily capacity target if it goes beyond this threshold****Question**

Can you estimate the probability to have less than 30 items per carton that you will receive every week?

*You can find the full code in this Github repository:**Link*## II. Central Limit Theorem

The Central Limit Theorem, defined as the** â€˜The Most Beautiful Theorem in Mathematicsâ€™** by my good friend Abdelkarim, establishes that when we add independent random variables, their normalized sum tends toward a normal

**distribution even when the original variables themselves are not normally distributed.**### 1. Definition

To simplify the comprehension, letâ€™s introduce some notations:

In our case, the total population is the entire scope of cartons received from the stores with a mean ** Âµ = 23 items per carton** and a standard deviation of

**Ïƒ = 7 items per carton.**If you take n samples of cartons X* n* (for instance a sample can be a batch of cartons received at a certain date) we have:

In other words, this means if we randomly measure the number of items per carton using n samples and we assume that observations are ** independent and identically distributed (i.i.d.)** the probability distribution of the sample means will closely

**.**

**approximate a normal distribution***Note: To ensure that we have independent and identically distributed observations we assume that the samples are built based on return batches coming from all stores in a scope covering 100% of the active SKU.*

### 2. Application

Then, can then assume that the average number items/case is following a normal distribution with a mean of ** 23 items per carton **and a

**standard deviation of 7 cartons.**What is the probability to have less 30 items per carton?

### 3. Probability to get <30 items per carton?

Probability to have less than 30 items/carton is 84.13%

**Code**

### 4. 95% probability to have less than k items per case?

Your KPI target is to have at least 95% of the returns processed the same day.

How many items do you need to assume to size your team for handling 95% of the expected workload? We have 95% of probability that X <= **34.51 items/carton**

If you size your team based on ** 35 items/carton** you will reach 95% of your target on average.

## III. Conclusion

This methodology gives you the possibility to size your team based on assumptions backed by powerful statistical tools.

This analysis can be performed several times per year, especially if the business is evolving (higher number of collections, e-commerce or new stores openings).