8.5 Inventory Control

8.5 Inventory Control

1. EOQ (Economic Order Quantity)

1. Basic Concept:

   • Optimal order quantity that minimizes total inventory costs.

   • Balances ordering costs and holding costs.

   • Assumptions:

     • Demand is constant and known.

     • Lead time is constant.

     • No quantity discounts.

     • No stockouts allowed.

     • Order arrives all at once.

2. Cost Components:

   • Ordering Cost ($S$): Cost to place and receive an order (paperwork, transportation, inspection).

   • Holding/Carrying Cost ($H$): Cost to hold one unit for one year (storage, insurance, capital, obsolescence).

     • Often expressed as percentage of item cost: $H = i \times C$ where $i$ = annual holding rate, $C$ = unit cost.

   • Purchase Cost ($C$): Cost per unit of item.

   • Total Annual Cost: $TC = \text{Purchase Cost} + \text{Ordering Cost} + \text{Holding Cost}$

3. EOQ Formula:

   • $EOQ = Q^* = \sqrt{\frac{2DS}{H}}$

     • $D$ = Annual demand (units/year)

     • $S$ = Ordering cost per order ($$/order)

     • $H$ = Holding cost per unit per year ($$/unit/year)

4. Derived Values:

   • Number of orders per year: $N = \frac{D}{Q^*}$

   • Time between orders (cycle time): $T = \frac{Q^*}{D}$ (in years) or $T = \frac{Q^*}{D} \times 365$ days

   • Total Annual Cost at EOQ: $TC_{min} = DC + \sqrt{2DSH}$

5. Extensions:

   • Quantity Discounts: Compare total costs at different price breakpoints.

   • Production Order Quantity (POQ): For items produced internally, not ordered externally.

     • $POQ = \sqrt{\frac{2DS}{H(1-\frac{d}{p})}}$ where $p$ = production rate, $d$ = demand rate.

2. Safety Stock

1. Purpose:

   • Buffer inventory to protect against uncertainties.

   • Prevents stockouts due to:

     • Demand variability.

     • Lead time variability.

     • Supply disruptions.

2. Factors Influencing Safety Stock:

   • Service Level: Probability of not having a stockout.

   • Demand Variability: Standard deviation of demand.

   • Lead Time Variability: Standard deviation of lead time.

   • Desired Service Level: Higher service level requires more safety stock.

3. Safety Stock Calculation Methods:

   • Basic Method (for constant lead time, variable demand):

     • $SS = z \times \sigma_d \times \sqrt{L}$

     • $z$ = service factor (from normal distribution)

     • $\sigma_d$ = standard deviation of daily demand

     • $L$ = lead time in days

   • General Method (variable demand and variable lead time):

     • $SS = z \times \sqrt{L\sigma_d^2 + d^2\sigma_L^2}$

     • $\sigma_L$ = standard deviation of lead time

     • $d$ = average daily demand

4. Service Level Considerations:

   • Common z-values:

     • 90% service level: $z = 1.28$

     • 95% service level: $z = 1.65$

     • 99% service level: $z = 2.33$

   • Service Level vs Cost: Higher service level increases holding costs.

5. Reorder Point with Safety Stock:

   • $ROP = (d \times L) + SS$

   • $d \times L$ = expected demand during lead time

   • SS = safety stock

3. ABC Analysis

1. Concept (Pareto Principle):

   • Small percentage of items account for large percentage of value.

   • 80/20 Rule: Approximately 80% of total inventory value comes from 20% of items.

2. Classification:

   • A Items:

     • Top 10-20% of items by annual dollar usage.

     • Account for 70-80% of total inventory value.

     • Management: Tight control, frequent review, low safety stock.

   • B Items:

     • Next 20-30% of items.

     • Account for 15-25% of total value.

     • Management: Moderate control, periodic review.

   • C Items:

     • Remaining 50-70% of items.

     • Account for 5-10% of total value.

     • Management: Simple control, large orders, high safety stock.

3. Analysis Procedure:

   1. Calculate annual dollar usage for each item: $Annual\ Usage = Unit\ Cost \times Annual\ Demand$

   2. Sort items in descending order of annual dollar usage.

   3. Calculate cumulative percentage of items and cumulative percentage of dollar usage.

   4. Plot Pareto curve and establish A, B, C boundaries.

   5. Apply appropriate inventory policies to each category.

4. Annual Dollar Usage Calculation:

   • For each item: $Value = C_i \times D_i$

   • Total inventory value: $TV = \sum_{i=1}^{n} C_i D_i$

   • Item percentage: $\frac{C_i D_i}{TV} \times 100\%$

5. Applications Beyond Inventory:

   • Supplier management.

   • Quality control (focus on critical defects).

   • Customer profitability analysis.

4. Forecasting

1. Forecasting Purpose:

   • Predict future demand for inventory planning.

   • Basis for production planning, purchasing, capacity planning.

   • Reduces uncertainty in inventory decisions.

2. Forecasting Methods:

   • Qualitative Methods:

     • Expert opinion, market research, Delphi method.

     • Used when little historical data exists.

   • Quantitative Methods:

     • Time series analysis.

     • Causal models.

     • Based on historical data patterns.

3. Time Series Components:

   • Trend (T): Long-term upward or downward movement.

   • Seasonality (S): Regular pattern repeating over periods.

   • Cyclical (C): Longer-term economic cycles.

   • Random (R): Unpredictable variation, noise.

4. Common Forecasting Techniques:

   • Naive Forecast: $F_{t+1} = A_t$ (next period = this period)

   • Simple Moving Average:

     • $F_{t+1} = \frac{A_t + A_{t-1} + ... + A_{t-n+1}}{n}$

     • Smooths random variations.

   • Weighted Moving Average:

     • $F_{t+1} = w_1A_t + w_2A_{t-1} + ... + w_nA_{t-n+1}$

     • $\sum w_i = 1$

   • Exponential Smoothing:

     • $F_{t+1} = \alpha A_t + (1-\alpha)F_t$

     • $0 < \alpha \leq 1$ (smoothing constant)

     • Simple form: $F_{t+1} = F_t + \alpha(A_t - F_t)$

5. Forecast Error Measurement:

   • Error: $e_t = A_t - F_t$

   • Mean Absolute Deviation (MAD): $MAD = \frac{\sum|e_t|}{n}$

   • Mean Squared Error (MSE): $MSE = \frac{\sum e_t^2}{n}$

   • Mean Absolute Percentage Error (MAPE): $MAPE = \frac{\sum|\frac{e_t}{A_t}|}{n} \times 100\%$

6. Forecasting for Inventory Control:

   • Determine appropriate forecasting method based on demand pattern.

   • Use forecast to calculate EOQ, safety stock, reorder points.

   • Regularly update forecasts based on actual demand.

   • Consider forecast error in safety stock calculations.