Ch. 20: Inventory Management
Inventory is the stock of any item or resource an organization uses; it helps to picture it as stacks of cash. An inventory system is the collection of policies and controls that determine which levels to hold, when to restock and what quantity to order. In manufacturing, inventory consists of raw materials, components, supplies, work-in-process and finished goods; in distribution there is inventory in transit and in warehouses. The models in this chapter suit items stocked at decoupling points (retail outlets, warehouses), whereas lower levels in the chain often rely on MRP.
Purposes and costs of inventory
Reasons to hold inventory: independence of operations, meeting variation in demand, flexibility in production scheduling, protection against variation in raw material delivery time, economic purchase order sizes, and domain-specific reasons (in-transit stock, anticipated price increases).
| Cost | Content | Effect |
|---|---|---|
| Holding (carrying) cost | Storage, handling, insurance, pilferage, obsolescence, taxes, opportunity cost of capital | Favors small inventories, frequent orders |
| Setup cost | Changing production to another product: materials, equipment setup, paperwork | Favors large lots (JIT aims to reduce it) |
| Ordering cost | Managerial and clerical cost of preparing orders and tracking them | Favors large, infrequent orders |
| Shortage cost | Stockouts (lost sales) and backorders | Favors extra stock |
Independent demand items have demand unrelated to other items (cars, forecast from the market). Dependent demand follows directly from the need for a higher-level item (2,000 wheels for 500 cars) and is simply computed.
Single-period model
Used for one-time purchases (newspapers, event T-shirts, fashion items, airline overbooking). Order up to the point where the probability of selling the last unit balances the costs.
P ≤ Cu / (Co + Cu), with Cu = cost of underestimating demand (lost profit per unit) and Co = cost of overestimating (loss per unsold unit). Order quantity = mean + z x σ, where z matches cumulative probability P.
Example: paper costs $0.20, sells for $0.50: Cu = 0.30, Co = 0.20, P = 0.6, z = 0.253. Demand mean 90, σ 10: stock 90 + 2.53, so 93 papers.
In the hotel example, Cu = $80 (empty room) and Co = $200 (paying another hotel), P = 0.2857, z = -0.566; with mean cancellations 5 and σ 3 the hotel should overbook 5 - 1.7, so 3 rooms.
Multiperiod systems: Q-model versus P-model
| Feature | Fixed-order quantity (Q-model) | Fixed-time period (P-model) |
|---|---|---|
| Order quantity | Q constant | q varies each time |
| When to order | When inventory position drops to reorder point R (event triggered) | When the review period T arrives (time triggered) |
| Recordkeeping | Every withdrawal or addition (perpetual) | Counted only at review |
| Size of inventory | Smaller | Larger (must also protect during the review period) |
| Typical items | Expensive, critical or important items | Lower-cost items, items ordered together from one supplier |
Inventory position = on-hand + on-order - backordered.
Economic order quantity
Assumptions: constant and uniform demand, constant lead time and price, holding cost based on average inventory, constant ordering cost, no backorders.
Total annual cost TC = DC + (D/Q)S + (Q/2)H.
Qopt = √(2DS / H).
Reorder point (certain demand) R = d x L.
D = annual demand, C = unit cost, S = cost per order or setup, H = annual holding cost per unit, d = average daily demand, L = lead time in days.
Example 20.2: D = 1,000, S = $5, H = $1.25, d = 1,000/365, L = 5: Q = √(2 x 1,000 x 5 / 1.25) = 89.4, so 89; R = 2.74 x 5 = 13.7, so 14.
At Qopt the annual ordering cost equals the annual holding cost ($56.18 versus $55.63 after rounding). The purchase cost DC does not affect Q when the price is constant.
Safety stock
Safety stock is inventory carried above expected demand. With the probability approach, choose z so that the probability of not stocking out equals the desired service level (z = 1 gives 84 percent, z = 1.64 gives 95 percent). It considers only the chance of a stockout, not the number of units short.
R = d x L + zσL, safety stock SS = zσL.
σL = √L x σd when daily demands are independent (the standard deviation of a sum is the square root of the sum of variances).
Example 20.4: d = 60, σd = 7, L = 6, S = $10, H = $0.50, 365 days: Q = √(2 x 21,900 x 10 / 0.5) = 936; σL = 7√6 = 17.15; R = 360 + 1.64 x 17.15 = 388.
q = d(T + L) + zσ(T+L) - I, with I = current inventory position and σ(T+L) = √(T + L) x σd.
Example 20.5: d = 10, σd = 3, T = 30, L = 14, 98 percent (z = 2.05), I = 150: q = 440 + 2.05 x 3 x √44 - 150 = 331 units.
Inventory turn and price breaks
Average inventory = Q/2 + SS. Inventory turn = cost of goods sold / average inventory value = DC / ((Q/2 + SS) x C).
The price-break model handles quantity discounts, with H = iC. Step 1: starting at the lowest price, compute the EOQ per price until one is feasible (falls in that price's range). Step 2: if that is the lowest price, it is optimal; otherwise compare its total cost with the total cost at each lower-price break quantity. Example 20.8: D = 10,000, S = $20, i = 20 percent, prices $5.00 / $4.50 / $3.90. EOQ at $3.90 is 716 (infeasible, needs 1,000+), at $4.50 is 667 (feasible). TC(667) = about $45,600 and TC(1,000) = $39,590, so order 1,000. Watch obsolescence and storage when buying beyond Qopt.
ABC classification and inventory accuracy
Following the Pareto principle (a few items carry most of the value), ABC classification ranks items by annual dollar volume (usage x unit cost): A items are roughly the top 15 percent of items with about 80 percent of value, B the next 35 percent with about 15 percent, C the last 50 percent with about 5 percent. A items get the tightest control (for example weekly ordering), C items the loosest. Unit cost alone does not decide the class, and critical items may be forced into A or B.
Inventory accuracy is the agreement between records and physical stock. It is protected with bar codes and RFID, locked storerooms, and cycle counting: counting items frequently instead of once or twice a year, triggered by low or zero balances, discrepancies, activity levels or ABC importance (for example $10,000+ annual usage every 30 days). Recommended tolerances: plus or minus 0.2 percent for A, 1 percent for B and 5 percent for C items.