Stochastic Modeling & Simulation Lecture 17 : Probabilistic Inventory Models part 2

Administrative Problem set 6 due Friday Projects: I’ll expect an update this Wednesday

Last time Derivation of the critical ratio (critical fractile)

(R,Q) Policies Multi-period inventory analysis with continuous review often results in an (R,Q) ordering policy. Q: the order quantity Last class we focused on a 1 period decision model with uncertain demain. R: the reorder point. Intuitively, R should be a function of what?

Deterministic Demand Inv Time Q=600 Order arrives Place Order Reorder Point=300 LT=3 wk Place Order Order arrives

Stochastic Demand Inv Time Order arrives Place Order LT=3 wk Place Order Order arrives LT=3 wk

(R,Q) Policies The reorder point (R) must depend on the demand that might be observed during the time between stock is reorder and its arrival (i.e., the lead time) The following is taken from example 12.7 in the scanned reading. Note: it assume that demand is Normally distributed. Normality is needed to make the math doable and easier, Assume: expected annual demand = 1200 units /year with standard deviation of 70 units. N(1200,70 2 ). Fixed ordering cost of $125 / order Holding cost of $8/unit per year Lead time = 1 week.

Demand During Lead Time If annual demand ~ N(1200,70 2 ), how much demand occurs during the time between reorder and arrival? Lead time is 1 week, or 1/52 of a year. Demand during the lead time is still unknown, but if we assume that the demand is stationary and not changing over time (along with a maybe 1 or 2 other assumptions)

Demand During Lead Time Given our assumptions that annual demand ~ N(1200,70 2 ) Demand during lead time ~ N(23 units, units) When should the company reorder? What is R in the (R,Q) policy? Probably not at 23. Why? Ordering at 23 units, implies that 50% of the time demand will exceed the current supplies and the company will stockout. If a stockoutor shortage is expensive to the company, then it should maybe keep more safety stock on hand. How much more? Optimal amount will be a function of the expense of a shortage. Suppose safety stock was 1 standard deviation of the lead time demand. What is the probability of a shortage?

Safety Stock How much safety stock to hold during the lead time is essentially the choice variable How much safety stock to hold should a function of the cost to the company of having a shortage – in addition to the additional holding costs, etc. Like the deterministic EOQ model, it’ll depend on total ordering and holding costs. Annual Ordering costs? Fixed cost * #Orders per year. = $125 * E(orders / year) = $125* 1200/Q.

Holding Costs Annual expected holding costs? Let h be the unit holding cost. How many units are held? Lowest expected inventory = safety stock = which we’ll often represent by by k standard deviations: Highest expected inventory = Q + safety stock = By symmetry of the distribution, on average the inventory will be (highest + lowest )/2 For a total holding cost of

Shortage costs How is the shortage or stockout cost determined? Sometimes it’s tricky (more on this in a few slides). Sometimes you can specify a $-amount. Say p = $10 / unit. Assumes that 2 storages is 2x as bad as 1 storage. If you just model is as lost sales, this is what you’re assuming. p * E(#shortages per cycle) * #cycles. E(#shortages per cycle)?

Expected backlogs It turns out that E(b) has some nice properties if we work with Normally distributed demand. Z = standard normal RV.

Spreadsheet Implementation See Ordering Cameras 1.xls

Shortage costs What if it’s too difficult to put a $-amount on the shortage cost? Or if we don’t think that 5 shortages = 5 * (cost of 1 shortage)? An alternative method deals with the fill rate, or service level. Fraction of demand that can be met from on-hand inventory. E.g. fill rate = 98%  at least 98% of the customer demand can be met from on-hand inventory. Under this specification, there is no unit shortage cost but we still have to keep track of expected # shortages per cycle, E(b)

Spreadsheet Implementation See Ordering Cameras 2.xls