Monte Carlo Simulation of Inventory Replenishment Levels

Stochastical solution of inventory replenishment problem.
business-analytics
python
lang-eng
Published

April 24, 2022

Stumbled upon the following problem from Camm’s Business Analytics:

Virtual Reality Goggle Inventory. Galaxy Co. sells virtual reality (VR) goggles, particularly targeting customers who like to play video games.

Galaxy procures each pair of goggles for 150 USD from its supplier and sells each pair of goggles for 300 USD. Monthly demand for the VR goggles is a normal random variable with a mean of 160 units and a standard deviation of 40 units. At the beginning of each month, Galaxy orders enough goggles from its supplier to bring the inventory level up to 140 goggles. If the monthly demand is less than 140, Galaxy pays 20 USD per pair of goggles that remains in inventory at the end of the month. If the monthly demand exceeds 140, Galaxy sells only the 140 pairs of goggles in stock. Galaxy assigns a shortage cost of 40 USD for each unit of demand that is unsatisfied to represent a loss-of-goodwill among its customers. Management would like to use a simulation model to analyze this situation.

  1. What is the average monthly profit resulting from its policy of stocking 140 pairs of goggles at the beginning of each month?
  1. What is the proportion of months in which demand is completely satisfied?
  1. Use the simulation model to compare the profitability of monthly replenishment levels of 140 and 160 pairs of goggles. Use a 95% confidence interval on the difference between the average profit that each replenishment level generates to make your comparison.

So, let’s tackle this problem, step by step.

1 Definition of Business Logic

  1. It logically follows from the problem definition that the monthly profit per unit is 150$.
  2. Monthly demand follows the normal distribution with parameters of \( = 160 \) and \( = 40 \), as showed in the KDE plot below. Do note that normal distribution is continuous, and here we are dealing with a discrete problem, so the generated values are rounded to zero decimal points. Also note that values below zero were transformed to zero, since the demand cannot be negative.

  1. The problem defines additional costs based on condition of demand satisfaction:
    1. If Demand \(\ge\) Supply: inventory is cleared, we incur cost of 40$ for each unit of unsatisfied demand.
    2. If Demand \(\lt\) Supply: demand is fully satisfied, we incur additional cost of 20$ for each unit that was not sold.

For all problems, single iteration set will consist of 20 000 samples.

2 Solution A

  1. What is the average monthly profit resulting from its policy of stocking 140 pairs of goggles at the beginning of each month?

So, based on predefined monthly replenishment level of inventory stock at level of 140 units, we could expect average monthly profit of 18,539$, and we can see that the average is pretty stable.

The plot above shows distribution of 10 000 averages. To be more clear, the following steps have been taken: 1. Generate 20,000 samples. 2. Calculate the mean. 3. Do this 10,000 times.

This approach is applied to Solution B and C also.

3 Solution B

  1. What is the proportion of months in which demand is completely satisfied?

So, if we are still in the domain of replenishing policy of 140 units, we can expect that the demand will be satisfied (Supply \(\ge\) Demand) in 30.4% of cases, on average.

Do note that this problem has analytical solution of 30.85%, since we are effectively searching for the area of normal distribution of Monthly Demand that is less or equal to 140:

4 Solution C

  1. Use the simulation model to compare the profitability of monthly replenishment levels of 140 and 160 pairs of goggles. Use a 95% confidence interval on the difference between the average profit that each replenishment level generates to make your comparison.

Defined as it is, solution to this problem will not lead to the optimal solution. So, I decided to test these levels for monthly replenishment levels between 100 and 500 pairs of googles.

This one is very interesting.

It seems that we can find 2 optimum points, and these optimum points can be selected based on the business strategy of the management, and/or it’s preferences and/or constraints of the organization.

The Mean Optimum strategy is based upon replenishment level of 206 units per month, which leads to expected profit of 22,623$.

On the other hand, based on the information contained in this problem, I would strongly advise going against the Mean Optimum and would heavily lean towards Risky Optimum. Why? The reasons are:

  1. For replenishment level of 206 units, we can expect around 30,000 USD profit on the high end, and around 10,000 USD profit on the low end.
  2. The Risky Optimum has lower average expected value (blue line), but not for much: we can expect around 22,000 USD of profit. The same stands for Low Expected value of profit: from 10k to 9.9k.
  3. But, the Risky Optimum sees a high jump in high section of Expected value regarding profit: from 30k that Mean Optimum has to whooping 35k.
  4. So, in effect, we are sacrificing minor drop in low and mean expected values of profit, for high-end gains of +5k, which is reasonable risk in my book.

Results of this simulation also prove my claim at the beginning of this section, that the original problem did not touch upon optimal levels of monthly inventory size. Levels between 140 and 160 are way below optimum and should be discarded, if profit is the only decision criteria.