Chapter 3 of 8

Inventory Optimization Under Uncertainty

3 sections · ~8 min read · 3 practice questions · part of Supply Chain Analytics

Learning objectives

  • Derive an economic order quantity and explain the shape of the total cost curve around it
  • Size safety stock when both demand and lead time vary, and state the assumptions this requires
  • Set a reorder point and translate a service level target into units and dollars

Two costs pulling in opposite directions

The order quantity decision balances two costs that move oppositely as the batch size changes. Ordering cost is incurred per replenishment regardless of size and covers purchase order administration, receiving, inbound handling, and any fixed freight or setup charge; total annual ordering cost equals annual demand divided by order quantity, times the cost per order, so it falls as the batch grows. Holding cost is incurred per unit per year and covers capital, storage, insurance, shrinkage, and obsolescence; average cycle stock is half the order quantity, so total annual holding cost rises linearly with batch size. Setting the derivative of their sum to zero gives the economic order quantity, the square root of twice annual demand times cost per order, divided by annual holding cost per unit. Two features of this result matter more than the formula. First, at the optimum the annual ordering cost exactly equals the annual cycle holding cost, which gives a free arithmetic check on any calculation. Second, the total cost curve is remarkably flat near the optimum, because the two components are trading off symmetrically. Total cycle-related cost at any quantity Q is the optimum multiplied by one half of the sum of Q over EOQ and EOQ over Q, so ordering fifty percent above the ideal quantity raises it by only about eight percent, and ordering twenty-five percent above raises it by about two and a half percent. This flatness is a licence to round the answer to a practical pallet, case, or truckload quantity without meaningful penalty, and it also means precision in estimating the cost per order is not worth agonising over.

Safety stock when demand and lead time both vary

Cycle stock covers expected demand between replenishments. Safety stock covers the surprise. The quantity that matters is not the variability of weekly demand alone but the variability of total demand over the replenishment lead time, because that is the exposure window. When lead time is a fixed constant, the standard deviation of lead time demand is the weekly standard deviation multiplied by the square root of the lead time in weeks. When lead time itself varies, a second term appears, equal to the mean weekly demand squared multiplied by the variance of the lead time. The combined standard deviation is the square root of the sum of those two terms. This composite formula assumes weekly demands are independent of one another, that demand and lead time are independent of each other, and that the weekly variability estimate is stable. Those assumptions are frequently violated, most obviously when a supplier's lead time stretches precisely because everyone's demand surged at once, in which case the true exposure is larger than the formula suggests. The second term is often the dominant one. Multiplying by a service factor, the number of standard deviations corresponding to the desired probability of not stocking out during a replenishment cycle, converts the exposure into units of safety stock. Note that this factor targets cycle service level, the probability of no stockout in a cycle, which is a stricter and different thing from fill rate, the fraction of demand met from stock.

From units to a decision

The reorder point is expected demand over the lead time plus safety stock, expressed in units, and it is the number the planning system watches. Every element must be in consistent time units: if the standard deviation is weekly, the lead time must be in weeks. The financial translation is what makes the number arguable. Multiply the safety stock in units by the annual holding cost per unit to get the annual carrying cost of the service level you selected, then repeat at a higher service factor to price the increment. Presenting service as a menu, for example that moving from roughly 95 percent to roughly 99 percent cycle service costs a specific number of additional dollars per year for this item, turns an abstract policy debate into a purchasing decision. It also exposes the diminishing returns clearly, since the service factor grows steeply as the target approaches 100 percent while the service gain shrinks.

Worked example

Problem

Marlow Instruments stocks a purchased sensor module. Annual demand is 24,000 units spread evenly across 52 weeks. The cost to place and receive one order is 120 dollars. Unit purchase cost is 18 dollars and the annual holding cost rate is 22 percent of unit cost. Weekly demand has a standard deviation of 90 units. Supplier lead time averages 2 weeks with a standard deviation of 0.5 weeks. Use a service factor of 1.65 for roughly 95 percent cycle service. Compute the economic order quantity, the standard deviation of lead time demand, safety stock, the reorder point, and total annual ordering plus holding cost.

Step by step

  1. Annual holding cost per unit = 18 dollars x 0.22 = 3.96 dollars per unit per year.
  2. EOQ numerator = 2 x 24,000 x 120 = 5,760,000. Divide by 3.96 to get 1,454,545.45. EOQ = square root of 1,454,545.45 = 1,206.05 units, round to about 1,206 units.
  3. Mean weekly demand = 24,000 / 52 = 461.54 units per week.
  4. Demand-variability term = lead time x weekly variance = 2 x (90 x 90) = 2 x 8,100 = 16,200.
  5. Lead-time-variability term = mean weekly demand squared x lead time variance = (461.54 x 461.54) x (0.5 x 0.5) = 213,017.75 x 0.25 = 53,254.44. Note this term is more than three times the demand term, so lead time reliability drives this item.
  6. Standard deviation of lead time demand = square root of (16,200 + 53,254.44) = square root of 69,454.44 = 263.54 units.
  7. Safety stock = 1.65 x 263.54 = 434.84 units, round to 435 units.
  8. Expected demand over lead time = 461.54 x 2 = 923.08 units. Reorder point = 923.08 + 434.84 = 1,357.92 units, round to about 1,358 units.
  9. Orders per year = 24,000 / 1,206.05 = 19.90. Annual ordering cost = 19.90 x 120 = 2,387.97 dollars.
  10. Annual cycle holding cost = (1,206.05 / 2) x 3.96 = 603.02 x 3.96 = 2,387.97 dollars, which matches the ordering cost and confirms the EOQ is correct.
  11. Annual safety stock holding cost = 434.84 x 3.96 = 1,721.98 dollars.
  12. Total annual ordering plus holding cost = 2,387.97 + 2,387.97 + 1,721.98 = 6,497.92 dollars.

Answer. EOQ = 1,206 units ordered about 19.9 times per year. Standard deviation of lead time demand = 263.54 units. Safety stock = 435 units and the reorder point = 1,358 units. Total annual ordering plus holding cost = 6,497.92 dollars, of which 1,721.98 dollars is the price of the roughly 95 percent cycle service level. The dominant insight is that 53,254 of the 69,454 total variance, about 77 percent, comes from lead time variability rather than demand variability, so this item's inventory is bought mainly to insulate the firm from an unreliable supplier, not from unpredictable customers.

Practice

Work each question before opening the solution.

  1. The supplier agrees to a firm 2-week lead time with no variability, leaving demand variability unchanged. Recompute the standard deviation of lead time demand, safety stock, and the annual dollar saving.

    Show solution for question 1

    With lead time variance zero, only the demand term survives: standard deviation = square root of 16,200 = 127.28 units. Safety stock = 1.65 x 127.28 = 210.01 units, down from 434.84 units, a reduction of 224.83 units. Annual saving = 224.83 x 3.96 = 890.34 dollars, and the reorder point falls from 1,358 to 923.08 + 210.01 = 1,133 units. This quantifies exactly what supplier reliability is worth on this item and gives the buyer a number to negotiate against.

  2. Warehouse handling constraints force the firm to order in full pallets of 2,000 units instead of the EOQ. Compute the annual cost penalty and comment on whether the constraint is worth fighting.

    Show solution for question 2

    At Q = 2,000: ordering cost = (24,000 / 2,000) x 120 = 12 x 120 = 1,440 dollars, cycle holding cost = (2,000 / 2) x 3.96 = 3,960 dollars, total = 5,400 dollars. At the EOQ the same two components total 2,387.97 + 2,387.97 = 4,775.94 dollars. The penalty is 5,400 - 4,775.94 = 624.06 dollars per year, about 13.1 percent. That is real but small, and it illustrates the flatness of the curve: a 66 percent increase in order quantity costs only 13 percent more. Fighting the pallet constraint is probably not worth management attention when a 890-dollar saving is available from lead time reliability instead.

  3. Leadership wants to raise cycle service from roughly 95 percent (factor 1.65) to roughly 99 percent (factor 2.33), keeping the current lead time variability. What is the additional safety stock, its annual cost, and the new reorder point?

    Show solution for question 3

    New safety stock = 2.33 x 263.54 = 614.05 units, an increase of 614.05 - 434.84 = 179.21 units. Additional annual holding cost = 179.21 x 3.96 = 709.67 dollars. New reorder point = 923.08 + 614.05 = 1,537.13, about 1,537 units. Note the shape of the trade: buying the last four percentage points of cycle service costs 709.67 dollars per year, while the 1,721.98 dollars already spent bought the move from 50 percent to 95 percent, because zero safety stock corresponds to a service factor of zero and therefore to roughly 50 percent cycle service, not to zero percent. So 1,721.98 dollars buys 45 points and the next 709.67 dollars buys 4, and the marginal price of service is rising steeply and should be justified against the contribution margin actually at risk in a stockout.