Develop a probabilistic inventory model using stochastic programming or simulation, incorporating probability distributions (e.g., Normal, Poisson) for demand and lead time variability.

Determine the Demand Distribution:

• Historical Data: Analyze historical demand data to understand the pattern of variability. Use statistical tools to estimate the distribution (e.g., normal, Poisson, or exponential) that best fits the demand data.
• External Factors: Identify and quantify the impact of external factors like seasonality, economic trends, and market conditions on demand. These can be incorporated into your demand distribution model.
• Estimate the Lead Time Distribution:
• Lead Time Uncertainty: Factor in lead time variability (if applicable) as it also affects the inventory model. Use historical data on lead times to estimate its probability distribution.

Determine the Desired Service Level:

• Service Level Goal: Decide on an appropriate service level (e.g., 95%, 99%). This represents the probability of meeting customer demand without running out of stock during a replenishment cycle.

. Select the Inventory Review System:

• Continuous Review (Q system): In this system, inventory levels are continuously monitored, and an order is placed when stock reaches the reorder point.
• Periodic Review (P system): Inventory is reviewed at fixed intervals, and the order size is adjusted to bring inventory back up to a target level.

Monte Carlo Simulation (Optional):

• Run simulations to test your inventory model under various demand and lead-time scenarios. This will help assess how well the model performs in maintaining the desired service levels under uncertainty.

7. Incorporate Costs:

• Consider holding costs, ordering costs, and stockout costs in your model. A common objective is to minimize total cost while achieving the desired service level.

8. Validate the Model:

• Test the model using historical data and fine-tune parameters as needed. You may adjust the safety stock or reorder point if the model fails to meet service levels or leads to excessive inventory.

In mathematical inventory models, historical demand data must be analyzed to understand the pattern of variation in demand by finding the demand rate and standard deviation, then finding the coefficient of variation for demand. Using the coefficient of variation, the type of demand can be determined, whether it is deterministic or probabilistic.

If the coefficient of variation is less than 20%, this means that the demand is deterministic . If the coefficient of variation is greater than 20%, this means that the demand is probabilistic and must be addressed using probability theory, as the probability distribution of the demand data is found.

*The type of lead time must be known in the place of application, whether it is deterministic or probabilistic, because it affects the inventory model. Using historical data, the type of lead time is known to estimate its probability distribution.

*Determining the level of service required by the decision maker in the place of application, usually estimated between (95 - 99%), which represents the probability of meeting customer demand without running out of stock during the replenishment cycle.

*The type of review of inventory levels is determined (continuous review or periodic review)

Continuous review: In this system, inventory levels are monitored continuously, and a supply request is submitted when the inventory reaches the reorder point.

Periodic review: Inventory is reviewed at fixed intervals, and the order volume is adjusted to restore inventory to a target level.

*Calculating inventory costs in your model:

1-Holding cost :It is the cost of keeping one unit for each Inventory cycle and includes the cost of place, insurance cost, examination and inspection cost

2-Setup cost :It is the cost of preparing the order or preparing the machines for each order, as it is calculated for each order and begins with the issuance of the purchase or production order and ends with the arrival of the materials to the stored. It includes costs issuing documents, transporting, unloading materials and arranging them in stores, communications, employee salaries and other administrative costs

3-Shortage cost: It is the represents the cost of loss that the company or institution will incur will incur due to not having inventory when it is needed, as well as the bad impact on the company's reputation.

The main goal of inventory models is to minimize total cost while achieving the required level of service

to meet customer demands.

*Intelligent techniques such as artificial neural network or deep learning can be used to predict demand based on historical demand data and then build a mathematical model for inventory based on the prediction results.

*Artificial intelligence algorithms such as genetic algorithm or swarm algorithms and others can be used to solve the probabilistic inventory model and improve the classical solution of the inventory model.