Operations management

Basic Economic Order Quantity Calculations

• Average inventory under optimal economic order quantity (EOQ) is expressed as $\frac{Q}{2}$.

• When average inventory is $20\,\text{units}$ and the annual morning cost (holding cost) per unit is $€2$, the total holding cost equals $2 \times 20 = 40$.

• At the optimal order quantity, inventory holding cost and ordering cost are equal.

EOQ Optimization vs. Fixed-Interval Ordering

• Evaluating an optimal EOQ against an arbitrary fixed-order schedule:

  • Total annual demand: $12{,}000\,\text{units}$.

  • Optimal EOQ policy: Yields approximately $219\,\text{units}$ per order, resulting in an optimal total inventory cost of $€438$.

  • Three-month ordering policy: Dividing $12\,\text{months}$ by $3\,\text{months}$ gives $4\,\text{orders}$ per year. The resulting order quantity is $Q = \frac{12{,}000}{4} = 3{,}000\,\text{units}$.

  • Total inventory cost under the three-month ordering policy is $€3{,}016$.

  • The cost penalty for using the arbitrary schedule is an extra $3{,}016 - 438$, which is roughly seven times higher than the optimized cost.

Quantity Discounts and Feasibility Evaluation

• Quantity discounts require incorporating purchasing cost ($P \times D$) and verifying if calculated EOQ values fall within valid price discount brackets.

• Baseline problem parameters:

  • Demand ($D$): $5{,}000\,\text{units}$

  • Ordering cost: $€49$

  • Holding cost ($H$): Proportional to price, specified as $2.2\text{ times the price}$ (and calculated with $4.2\text{ times the price}$, as holding costs decrease with lower prices)

• Price bracket evaluation:

  • Price $€5$ ($0 - 999\,\text{units}$): Calculated EOQ is $700\,\text{units}$. This quantity falls within the bracket and represents a feasible, valid solution.

  • Price $€4.8$ ($1{,}000 - 1{,}999\,\text{units}$): Calculated EOQ is $714\,\text{units}$. Because $714\,\text{units}$ is outside the qualifying range, the lowest valid quantity to secure this discount is $1{,}000\,\text{units}$.

  • Price $€4.75$ ($\ge 2{,}000\,\text{units}$): The unconstrained EOQ is below the qualifying threshold; evaluating the minimum bracket boundary of $2{,}000\,\text{units}$ yields savings of almost $€1{,}000$ compared to the base scenario.

Questions & Discussion

• Scope of Average Inventory:

  • Question: Does average inventory refer to the average inventory of a single month?

  • Response: Average inventory applies across the entire period (whether one month or one year), calculated directly as $\frac{Q}{2}$.

• Derivation of Order Quantity from Order Frequency:

  • Question: Why is the demand value of $12{,}000$ divided when calculating the order quantity?

  • Response: The number of orders is defined by the formula $N = \frac{D}{Q}$, which rearranges algebraically to $Q = \frac{D}{N}$.

• Historical Algorithmic Capability:

  • Discussion noted that during its first two years, Chezip meeting failed to solve this inventory problem under the assumption that essential data was missing, though it has since solved it successfully.