10 - inventory mgment 3

Question 1

what are the two periodic inventory models with uncertainty that we look at?

Answer 1

r,S and s,Q models

Question 2

what are the two main categories of periodic models depending on when to order?

Answer 2

1. Order when the inventory level reaches a certain inventory level, “s”

Applicable when you constantly monitor inventory; if you have accurate and real time inventory data, and ordering process is largely automatic

2. Order in constant intervals “r”, e.g., order at the end of each week

Used when you need to conduct periodic inventory counting; other logistical concerns may be involved. useful with supplier agreements, or part of larger system and management wants to serve all branch retailers at same time.

Question 3

what are the two strategies for order qty/lot size in multi period models?

Answer 3

1. Ordering a fixed lot size Q
2. Ordering up to a certain level S

Question 4

what is the (r,S) model of inventory management?

Answer 4

constant intervals, to a certain level

inventory reviewed in steady periods

depending on available inv, order is placed to increase the inv level up to S

in doing so, length of the review period and lead time, and the expected demand in that period is considered.

Moreover, a desired service level is considered for the calculation of the safety stock.

review epochs = when orders placed.
time between review epochs = review periods

Question 5

what are the assumptions of the (r,S) model?

Answer 5

Review periods are fixed, so unlike EOQ model, the frequency of ordering in this model is not a decision variable.

Instead, the tradeoff is between holding cost of inventory and shortage cost.

Unmet demand « waits » but customers receive compensation b per unit/time for waiting

Inventory at the end of the review period can be re-used in the next period. However, costs for keeping inventory h per unit accrue

Question 6

what is shortage cost in multi period models?

Answer 6

shortage cost = contractual obligation, if you don’t supply on time sometimes there is a genuine penalty fee (not just opportunity cost). can include both elements or just one, depends on context.

(holding, shortage and ordering costs generally the main 3 costs considered in real world)

Question 7

what is the policy under the (r,S) model?

Answer 7

Place an order Qt at each review epoch, knowing that the order arrives after L time units (lead time)

Question 8

what is the order quantity under the (r,S) model?

Answer 8

The order quantity should be chosen such that your current inventory and your open orders adds up to S

Question 9

what is the optimal service level in the (r,S) model?

Answer 9

with unit holding cost h per unit time, and backlog/shortage/lost sale cost b per unit time, the optimal service level, similar to the newsvendor mode, balances b and h:

beta(optimal) = b/(b+h)

b similar to underage, h similar to overage

Question 10

what is the (s,Q) model?

Answer 10

at certain inv level, place fixed order size

Reorder point, s = Inventory level at which we should place fixed order Quantity, Q

lead time assumed to be fixed, but demand during lead time - and in general - is uncertain

Question 11

what is the policy under the (s,Q) model?

Answer 11

if inv level is at s, we order Q

in absence of shortage cost, Q = EOQ = sqrt(2KD/H), where k is fixed ordering cost, d is annual demand and h is annual unit holding cost

Question 12

how is s calculated in the (s,Q) model?

Answer 12

demand during lead time normally dist with mean muL, want s to satisfy

P(D ≤ s) = beta — optimal or imposed service level.

if you want higher service level, will need higher s and safety stock.

safety stock = difference between average demand during lead time and s

Question 13

what is assumed when finding the appropriate level of safety inventory in the (s,Q) model?

Answer 13

Demand per period is normally distributed, with mean mu and standard deviation sigma.

Inventory policy is continuous review with a lot size Q ordered when the inventory at hand declines to the reorder point s.

The replenishment lead time is fixed and = L periods

Reorder point (s) = pipeline inventory(mu*L) + safety stock (ss)

safety stock (ss) = s - mu*L

Question 14

what is the cycle service level (CSL)?

Answer 14

Fraction of replenishment cycles that end with all customer demand being met

Probability of not having a stockout in a replenishment cycle

Question 15

how can you find the reorder point s when given a target cycle service level?

Answer 15

beta = Prob(demand during lead time ≤ s)

s = muL + ss

= muL + Zbeta*sdL

Question 16

how can safety inventory be reduced?

Answer 16

reduce the cycle service level - reduces Zbeta. not linear so can be very useful, but may not have control of this

reduce lead time - lowers sqrt(L). does require new supplier.

reduce uncertainty in demand/improve forecasting - reduces sigma, not always possible

Question 17

what is the difference between inventory pooling/decentralising stock in a multi period setting?

Answer 17

decentralised: if warehouses are all homogeneous (same sd)

total safety inv = Zbetasqrt(L)k*sigma

centralised:
aggregate demand normally dist. with sd sigmaCD

total safety inventory = Zbetasqrt(kL)sigma

Question 18

what is the effect of inventory pooling in a multi period setting?

Answer 18

you get to square root the k in the equations - if lead time is large this can have big benefit as this is the multiplier. remember this reduces safety stock, so stock that is just sat around anyway, so can remove significant deadweight costs.

aggregation reduces demand uncertainty, and therefore the required safety inventory.