Economic Order Quantity (EOQ) Model Quiz

A system where replenishment orders are placed when the inventory position (IP) reaches a predetermined level called the reorder point (s), and the fixed order quantity (Q) is ordered.

The inventory position (IP) is the sum of the on-hand inventory plus any outstanding orders minus any backorders.

The Economic Order Quantity (EOQ) is the optimal order quantity that minimizes the total inventory holding cost and ordering cost.

The assumptions of the basic EOQ model include no undershoot (IP = s when order is placed), unbiased demand forecast, deterministic lead time, normally distributed forecast errors, complete backorders, and negligible backorder probability after order arrives.

Total cost per year is the sum of order cost and cycle stock cost.

The probability of having enough inventory to meet demand during a replenishment cycle is called cycle service level (P1), or the probability of no stockout in a cycle.

Safety stock (SS) is the extra inventory held to protect against unexpected demand fluctuations during the lead time.

Safety stock is calculated as the safety factor (k) multiplied by the standard deviation of lead time demand (σL).

The probability that the backorder at stage 2 will be equal to x, where x is a positive integer. It is calculated based on the probability of demand during lead time, where demand follows a Poisson distribution with a rate of lambda times lead time.

The probability that the inventory at stage 2 will be equal to x, where x is a positive integer. It is determined by the probability of demand being less than or equal to the available stock at stage 2.

The expected value of backorder at stage 2, calculated by summing the product of each possible backorder value and its corresponding probability.

The variance of backorder at stage 2, obtained by subtracting the square of the expected backorder from the expected value of the squared backorder.

The expected value of the squared backorder at stage 2, calculated by summing the product of each squared backorder value and its corresponding probability.

A serial system where there are two stages in the supply chain, with the second stage being the final stage closest to the customer.

In a two-stage serial system with installation stock, the backorder at stage 2 is calculated as the difference between demand at stage 2 and the available stock at stage 2, taking into account the positive difference.

In a two-stage serial system with installation stock, the inventory at stage 2 is calculated as the difference between the available stock at stage 2 and demand at stage 2, taking into account the positive difference.

The expected shortage per replenishment cycle (ESPRC) is calculated as the probability of running out of stock multiplied by the lot size. It can be estimated using the formula: ESPRC = (1 - P2) * lot size, where P2 is the probability of a stockout. The ESPRC represents the average amount of demand that is not met during a replenishment cycle due to stockouts.

The expected value of a random variable Z, representing the demand during the review period R, can be approximated as the mean of the demand during R squared divided by twice the expected demand during R: E[Z] ≈ E[DR]^2 / (2E[DR]). This formula helps estimate the average demand during the review period based on the expected demand during R.

The expected order size is determined by a simple formula: S - s + (σR^2 + x̂R^2) / (2x̂R), where s is the reorder point, S is the maximum inventory level, σR is the standard deviation of demand during the review period, and x̂R is the expected demand during the review period. The formula considers the variability of demand and ensures sufficient inventory to meet demand during the review period.

The reorder point (s) in a periodic review system (R, s, Q) is calculated as the expected demand during the lead time plus the expected demand during the review period plus a safety factor multiplied by the standard deviation of the total demand: s ≈ E[DL] + E[Z] + k * SD[DL + Z]. This formula helps ensure enough inventory is readily available to meet demand during both the lead time and the review period.

The variance of the demand during the review period (Var[Z]) can be approximated by a formula that considers the expected demand during R and its variability: Var[Z] ≈ E[DR^3]/(3E[DR]) - 1/4 * (E[DR^2]/E[DR])^2 - 1/12. This formula helps assess the risk of demand fluctuations impacting the order size and the likelihood of stockouts.

The standard deviation of the sum of lead time demand (DL) and demand during the review period Z (SD[DL + Z]) is calculated by taking the square root of the sum of the variances: √Var[DL] + Var[Z]. It provides a measure of the total variability in demand during the lead time and the review period.

Multi-item coordination is a strategy used in inventory management to optimize the ordering and replenishment process for multiple items that share common characteristics, such as location, supplier, or transportation mode. By coordinating the ordering, we can achieve cost savings through economies of scale and improved efficiency.

An (R, s, Q) system is a type of inventory management system where inventory is reviewed periodically (R) and a fixed order quantity (Q) is placed whenever the inventory level drops below the reorder point (s). This system aims to balance the need for inventory and the cost of holding it by replenishing the stock at regular intervals.

The cumulative distribution of demand during time period L, where 𝐷𝑗′ represents the echelon demand from the first period to the j-th period.

The minimum value of x that satisfies the condition that 𝐹𝐷1′(𝑥) is greater than or equal to the given fraction, which represents the required service level.

In this context, it is a newsvendor problem with a holding cost and backorder cost, designed to find a lower bound for the optimal stocking level at stage j.

The expected value of a random variable Z1, which is the sum of two other random variables B2 and D1, is equal to the sum of the expected values of those two variables.

The variance of a random variable Z1, which is the sum of two other random variables B2 and D1, is equal to the sum of the variances of those two variables.

The optimal stocking level at stage 2, determined using Shang and Song's heuristic.

A method to calculate the optimal stocking levels for each stage in a multi-stage supply chain, taking into account the demand distribution and service level requirements.

If a random variable has a negative binomial distribution, its first two moments (mean and variance) can be used to determine the parameters of the distribution: n1 (number of failures) and p1 (probability of success).

A negative binomial distribution describes the probability of getting a certain number of successes (x) before a fixed number of failures (n1) is reached, with each trial having a probability of success (p1).

The Gamma function is a generalization of the factorial function to complex numbers. It is defined for all complex numbers except non-positive integers.

The expected value of the backorders (I1) depends on the probability mass function of the backorders, calculating the expected number of backorders over all possible backorder quantities.

The total cost of the system includes the sum of holding costs, backorder costs, and transportation costs across all stages, taking into account the expected value of the inventory at each stage and the expected number of backorders.

The reorder points (Sj) for each stage are not independent. Setting higher reorder points in one stage may necessitate lower reorder points in other stages to maintain an optimal balance.

The order-up-to level for a warehouse in a guaranteed service model, calculated as the sum of mean demand during the net replenishment time and safety stock.

The difference between the promised service time by upstream to a warehouse (sjin) and the promised service time by the warehouse to its downstream (sjout).

The time between when a warehouse places an order from upstream and when it receives it. Includes lead time and service time.

The probability of a warehouse having enough inventory to meet demand during a replenishment cycle, expressed as a percentage.

A mathematical representation of a guaranteed service model optimization problem, where the objective is to minimize the total inventory holding costs subject to service level constraints.

The service level constraint applies to all warehouse systems, ensuring that the promised service time to downstream customers is always less than or equal to the total lead time plus the incoming service time.

The service level constraint for serial systems, where the incoming service time is simply the outgoing service time of the previous stage in the supply chain. This helps to ensure smooth flow of goods within the system.

The service level constraint for distribution systems, where the incoming service time is the outgoing service time of the stage that supplies that specific warehouse. This ensures consistent service levels across the entire distribution network.