Inventory Policy: Method

EOQ Family, (Q, r) and (R, S) Policies, Newsvendor and ABC

What an inventory policy decides

Stock exists because demand and supply do not arrive at the same moments. Holding it costs money (capital, space, insurance, obsolescence), ordering it costs money (paperwork, transport, setups) and running out costs money (lost sales, backorder handling, lost goodwill). An inventory policy answers two questions for one item: how much to order at a time, and when. The deterministic models answer the first with a lot size that balances ordering against holding; the stochastic models answer the second with a reorder point or an order-up-to level that balances holding against the risk of a shortage; the single-period model answers both at once for an item that cannot be reordered. All of the tool's models take one period as the time basis: the demand, the holding and shortage cost rates, the lead time and the review period are all per that period, which may be a year, a month, a week or a day.

Economic order quantity

Harris (1913) set the problem: demand at a constant rate D per period, a fixed cost K per order, a holding cost h per unit per period, replenishment that arrives all at once when the stock reaches zero. A lot Q is ordered D/Q times a period, so the ordering cost is DK/Q; the stock runs from Q to 0 linearly, so the average inventory is Q/2 and the holding cost is hQ/2. The total variable cost DK/Q + hQ/2 is minimized where the two terms are equal, at Q* = √(2DK/h), with the minimum cost √(2DKh), the cycle Q*/D and D/Q* orders per period. The purchase cost Dc does not depend on Q and is listed separately. The total is flat near the optimum: for any lot Q, TVC(Q)/TVC(Q*) = ½(Q/Q* + Q*/Q), so a lot 25 % off costs 2.5 % more and a lot twice or half the optimum 25 % more; rounding to a package quantity is cheap. When the holding cost is given as a carrying rate (interest plus storage as a fraction of the unit cost per period), h = (rate) × c.

Production lot (EPQ)

When the lot is made rather than bought, it arrives at the production rate P per period while demand D continues, so the stock climbs at P − D during the run of length Q/P and peaks at Q(1 − D/P). The average inventory is half that peak, and the lot that balances setups against holding is Q = √(2DK/(h(1 − D/P))), larger than the EOQ because less of the lot is ever in stock at once (Taft 1918). The model needs P > D; as P grows without bound it returns to the EOQ.

Planned shortages

If customers wait for backorders at a cost p per unit per period of waiting, a cycle can run the stock down past zero. With S the maximum inventory and Q − S the maximum shortage, the holding cost is hS²/(2Q) and the shortage cost p(Q − S)²/(2Q); minimizing over both gives Q* = √(2DK/h)·√((p + h)/p) and S* = √(2DK/h)·√(p/(p + h)) (Hillier and Lieberman §18.3). The lot is larger and the stock lower than without shortages, and the total variable cost falls to √(2DKh)·√(p/(p + h)). As p grows the model returns to the basic EOQ. The tool's default case is the TV speaker example of Hillier and Lieberman, whose printed results (25,298 every 3.162 months; 28,540 and 22,424 with shortages) it reproduces.

All-units quantity discounts

Suppliers often price by quantity: every unit of a lot of at least q_j costs c_j. The total cost per period at price c_j is Dc_j + DK/Q + hQ/2, a curve of the EOQ shape shifted down by the price step, so the cheapest lot is found by taking each price in turn: compute its EOQ (with h from the carrying rate at that price, or the entered h), move the lot up to the range's minimum if the EOQ falls below it (the curve rises on either side of its minimum, so the nearest feasible point is best), discard it if the EOQ lies above the range, and compare the total costs of the surviving candidates. A larger lot at a lower price wins when the saving on the purchase outweighs the extra holding cost; the tool draws the stepped total-cost curve so the comparison can be seen.

Lead-time demand

Between placing an order and receiving it the lead time L elapses and demand continues; what the stock must cover is the demand during that time, a random quantity. Entered directly, it is a mean μ and a standard deviation σ. Built from the demand per period (mean D, standard deviation σ_d) and a lead time with mean L and standard deviation σ_LT, the mean is DL and the variance Lσ_d² + D²σ_LT², the first term from the period-to-period variation of demand, the second from the variation of the lead time itself. The models assume this demand is normal, adequate when the mean is several times the standard deviation; for slow movers a Poisson or empirical treatment is more appropriate and is not offered here.

Continuous review: the (Q, r) policy

The stock is watched continuously and a lot Q is ordered whenever the inventory position (on hand plus on order less backorders) falls to the reorder point r. The lot comes from the EOQ; the reorder point is the mean lead-time demand plus a safety stock, r = μ + zσ, and the policy is defined by how z is chosen (Nahmias Chapter 5; Silver, Pyke and Peterson Chapter 7):

• Cycle service level α, the probability that the demand during a lead time does not exceed r: z = Φ⁻¹(α). Easy to state, but it counts cycles with a stockout regardless of how many units are short, and with large lots few cycles occur.
• Fill rate β, the fraction of demand met from stock. The expected units short in a cycle are ESC = σG(z), where G(z) = φ(z) − z[1 − Φ(z)] is the standard normal loss function (tabulated by Silver, Pyke and Peterson, Appendix B: G(1.0) = 0.0833, G(2.0) = 0.0085); the fill rate is 1 − ESC/Q, so the required z solves G(z) = (1 − β)Q/σ. The fill rate is what customers experience and is the measure to prefer.
• Shortage cost p per unit short. The cost per period, DK/Q + h(Q/2 + r − μ) + pD·ESC/Q, is minimized by the pair of conditions 1 − Φ(z) = Qh/(pD) and Q = √(2D(K + p·ESC)/h) (Hadley and Whitin 1963). Starting from Q₀ = EOQ the two are iterated: r from the first, ESC from r, Q from the second, until they settle, usually in three or four steps. The tool prints each step; the textbook procedure rounds to whole units at every step, so its final pair can differ from the continuous solution by a unit.

Whichever policy is selected, all three are tabulated with the same service measures (cycle service level, ESC, fill rate, units short per period, average inventory Q/2 + safety stock) and the cost split, so the price of a service target can be read against the cost-based optimum. A reorder point entered by hand is evaluated the same way.

Periodic review: the (R, S) policy

When the stock is counted only every R periods, the order placed at a review must last until the next order arrives, R + L periods later, so the order-up-to level is S = D(R + L) + zσ_R+L with σ_R+L = √((R + L)σ_d² + D²σ_LT²) and z = Φ⁻¹(α) for the chosen service level over the protection interval (Chopra and Meindl Chapter 12). The mean order is DR and the average inventory DR/2 + zσ_R+L; the safety stock is larger than under continuous review because it protects a longer interval, the price paid for reviewing less often. When the lead-time demand is entered directly, its variance is scaled in proportion to the time, σ_R+L = σ_L√((R + L)/L).

Newsvendor

For an item ordered once for a single selling season (newspapers, fashion, seasonal parts), each unit short costs the margin plus any goodwill penalty, c_u = price − c + penalty, and each unit left over costs the unit cost less the salvage value, c_o = c − salvage. The expected profit is maximized at the smallest Q with P(demand ≤ Q) ≥ c_u/(c_u + c_o), the critical ratio (Arrow, Harris and Marschak 1951). For normal demand Q* = μ + zσ with z = Φ⁻¹(ratio); the expected lost sales are σG(z), the expected sales μ − σG(z), the expected leftovers Q − sales, and the expected profit is price × sales + salvage × leftovers − c × Q − penalty × lost sales. With an empirical demand table the same quantities are sums over the table, and the tool lists the expected profit of every candidate quantity. A high margin pushes the ratio and the order above the mean demand; a low margin pulls them below it.

ABC classification

A few items usually account for most of the money moving through a stockroom. Ranking items by the value of their usage per period (usage × unit cost) and accumulating the percentages gives the Pareto curve; class A is the top of it (typically 20 % of the items and 70–80 % of the value), class C the long tail, B between (Dickie 1951). The tool marks an item A while the cumulative value before it is below the A cut-off, so the item that crosses the line is still A, B until the B cut-off, and C beyond; the cut-offs are inputs because the line is a management choice, not a statistic. Class A items deserve individual forecasts, frequent review and a measured σ for their safety stock; class C items are run on simple rules with large lots so that attention goes where the money is.

Reading the results

Every equation is printed with its substituted numbers. The cost curves show how flat the total is near the optimum: use them before insisting on an exact lot. The (Q, r) comparison table shows what each service target costs against the cost-based policy; if the cost-based reorder point gives a fill rate far below what customers tolerate, the shortage cost entered is too low. The periodic-review safety stock should be compared with the continuous-review one for the same item to see what the review interval costs. The newsvendor profit curve falls faster on the side of the larger unit cost; the entered quantity is placed on it. The models assume stationary demand, a single item and no capacity limits; seasonal demand, joint replenishment and multi-echelon stock are outside their scope.