EOQ and reorder point worked example: how sensitive is total cost to the order quantity?

EOQ balances ordering cost, which falls as you order larger lots, against holding cost, which rises. Annual cost = (D / Q) × S + (Q / 2) × H, and the minimum is at Q* = sqrt(2 × D × S / H) = sqrt(2 × 10,000 × 40 / 2) = sqrt(400,000) = 632 units.

At that quantity you place 10,000 / 632 = 15.8 orders a year. Ordering cost is 15.8 × 40 = 632 and holding cost is 316 × 2 = 632, total 1265 per year. The two parts are always equal at the optimum, which is a useful check on the arithmetic.

The cost curve is very flat near the minimum. The ratio of total cost at any Q to the minimum is ½ × (Q / Q* + Q* / Q). For a lot of 500: ½ × (0.79 + 1.26) = 1.03, only 3 percent more. For 1000: ½ × (1.58 + 0.63) = 1.11, 11 percent more, about 135 a year.

So rounding to a convenient pack size, pallet quantity or price-break quantity costs little, and there is no point calculating EOQ to the last unit, especially since S and H are estimates. Here a lot of 500 is the better of the two on these costs.

The reorder point covers demand during the lead time plus safety stock. Average daily demand is 10,000 / 250 = 40 units, so lead-time demand is 40 × 5 = 200 units. Safety stock = z × σ_d × sqrt(L) = 1.645 × 12 × sqrt(5) = 44 units for a 95 percent probability of not running out during a replenishment cycle. Reorder point = 244 units.

z comes from the normal distribution for the chosen cycle service level: 1.28 for 90 percent, 1.645 for 95 percent, 2.33 for 99 percent. Note how fast the stock grows at the top end: 99 percent needs about 62 units, over 40 percent more than 95 percent does.

The formula carries assumptions that are worth stating in the assignment: demand is steady and known on average, the whole order arrives at once, the price does not depend on quantity, and the ordering and holding costs are truly variable. Holding cost in particular should be the cost of capital plus storage, insurance and obsolescence per unit, usually expressed as a percentage of unit cost per year; it is often underestimated.

With quantity discounts, compute the total annual cost including purchase cost at the EOQ and at each price-break quantity and pick the lowest. With lumpy or seasonal demand, EOQ is a poor fit and period-based lot sizing is used instead.

Adding to the point about safety stock: the square-root formula above assumes the lead time is fixed. If the lead time varies too, the combined standard deviation is sqrt(L × σ_d² + d² × σ_L²), where d is average daily demand and σ_L is the standard deviation of lead time in days. With σ_L of just 1 day the second term is 40² × 1 = 1600 against 5 × 144 = 720 for the first, so the supplier's unreliability contributes more than demand variation, and safety stock rises to 1.645 × sqrt(2320) = 79 units.

Reducing lead-time variability is often the cheapest way to cut safety stock.