1. Inventory Control
IME 451, Lecture 3

2. Economic Order Quantity
Harris (1913) developed this basic, widely used model to find economic lot sizes
Balancing inventory holding costs against setup (or order) costs
Assumptions
Production is instantaneous, no capacity constraint
Delivery is immediate, no time lag
Demand is deterministic, no uncertainty
Demand is constant over time
A production run incurs a fixed setup cost
Products can be analyzed individually (single product only or no interactions such as shared resources or machines)

3. EOQ Variables D = demand rate (in units per year)
c = unit production cost, not counting setup or inventory cost (in dollars per unit)
A = fixed setup (ordering) cost to produce (purchase) a lot (in dollars)
h = holding cost (in dollars per unit per year); if holding costs are only due to interest then h=ic where i is the annual interest rate
Q = lot size (in units); this is the decision variable

4. EOQ Derivation Total annual cost Y(Q)
Find the minimum Y(Q) by setting derivative w.r.t. Q equal to 0
Check that the second derivative is positive for any positive Q (convex function, so Q* is a min, not a max)
Solve first derivative for Q*

5. Problems with EOQ
How realistic are assumptions (Instantaneous production? Deterministic demand?)
Setup costs may be difficult to estimate, especially in production environments rather than purchasing systems
Average number of lots per year, F
Total inventory investment, I
Time between orders, T

6. Sensitivity of EOQ Models
Holding and setup costs are fairly insensitive to lot size
Errors caused by restricting lot sizes to powers of 2 are minimal (no more than 6%)
Powers of 2 ordering can facilitate sharing truck resources (one week, two weeks, four weeks…)
Extensions involve non-instantaneous production (economic production lot model), backorders, major and minor setups, and quantity discounts

7. Dynamic Lot Sizing (Wagner Whitin)
Notation
t = a time period = 1, 2, …, T where T is the planning horizon
Dt = demand in period t (in units)
ct = unit production cost (in dollars per unit)
At = setup cost to produce a lot in period t (in dollars)
ht = holding cost to carry a unit of inventory from period t to period t+1 (in dollars per unit per period)
It = inventory (in units) leftover at the end of period t
Qt = lot size for period t (in units); there are T decision variables, one for each period

8. Wagner Whitin Procedure
Qt will be 0 or will be Dt, Dt+Dt+1 , Dt+Dt+1+Dt+2 ...
Produce nothing or produce exactly enough to cover the current period plus some integer number of future periods
Produce for the first period in the first period
For each subsequent period, decide whether it is more economical to produce that period’s demand in the current period or any previous period
Follow example in book, pp

9. Models for Uncertain Demand
Finding statistical reorder points to account for randomness in demand
News Vendor – single replenishment; vendor buys paper at start of day and discards any leftover at end of day
Base Stock – replenish inventory one unit at a time but carry base stock to cover lag time
(Q,r) model – when inventory reaches or falls below level r, order a quantity of Q items

10. News Vendor Model Assumptions Notation
Products are separable, consider 1 at a time
Planning is for a single period only
Demand is random
Deliveries are made in advance of demand
Costs of overage or underage are linear
Notation
X = demand (in units), a random variable with mean m and standard deviation s
G(x) = P(X<=x) = c.d.f. of demand
cs = cost (in dollars) per unit of shortage
co = cost (in dollars) per unit of overage
Q = production or order quantity (in units); this is the decision variable

11. News Vendor Equations
To balance overage vs shortage costs, choose order quantity Q*
Assume that G is normal, where F is the cdf of the standard normal function
Find z in a standard normal table (Table 1 at end of text)
Solve for Q*

12. Base Stock Model Assumptions Notation
Products can be analyzed individually
Demands occur one at a time
Unfilled demand is backordered
Replenishment leadtimes are fixed and known
Replenishments are ordered one at a time
Notation
l = replenishment leadtime (in days), assumed constant
X = demand during replenishment leadtime (in units)
G(x) = P(X<=x) = c.d.f. probability demand during replenishment leadtime is less than or equal to x
q = E[X] = mean demand (in units) during leadtime l
s = E[X] = standard deviation of demand (in units) during leadtime l

13. Base Stock Model Notation (continued)
h = cost to carry one unit of inventory for one year
b = cost to carry one unit of backorder for one year
r = reorder point (in units)
R = r + 1 = base stock level (in units)
s = r - q = safety stock level (in units)
S( R ) = fill rate or service level, fraction of orders filled from stock as a function of R
B( R ) = average outstanding backorders
I( R ) = average on-hand inventory
Inventory position = on-hand inventory – backorders + orders

14. Base Stock Equations
For this model, at all times inventory position = R
Service level, S( R ) = G (R – 1)
Backorders are 0 if x < R
Backorders are x-R if x>=R
Expected backorder level B( R )
I ( R ) = R – q + B( R )
Use table 2.5 to find fill rates

15. (Q,r) Model Assumptions
Similar to base stock except
There is a fixed cost for each replenishment order, OR
There is a constraint on the number of orders per year
Decide how much safety stock to carry to cover leadtimes AND what quantity to order

16. (Q,r) Notation D = expected demand per year (in units)
l = replenishment leadtime (in days)
X = demand during replenishment leadtime (in units), random variable
q = E[X] = Dl/365 = expected demand (in units) during leadtime l
s = E[X] = standard deviation of demand (in units) during leadtime l
G(x) = P(X<=x) = c.d.f. probability demand during replenishment leadtime is less than or equal to x
A = setup cost per replenishment (in dollars)
c = unit production cost (in dollars per unit)
h = cost to carry one unit of inventory for one year
k = cost per stockout
b = cost to carry one unit of backorder for one year
r = reorder point (in units)
Q = replenishment quantity
s = r - q = safety stock level (in units)
F (Q,r) = order frequency, as a fuction of Q and r
S( Q,r ) = fill rate or service level, fraction of orders filled from stock as a function of Q and r
B( Q,r ) = average outstanding backorders
I( Q,r ) = average on-hand inventory

17. (Q,r) Equations
Replenishment quantity Q affects cycle stock, inventory that is held to avoid excessive replenishment costs (like EOQ)
Reorder point r affects safety stock, inventory held to avoid stockouts (like Base Stock)
Either minimize: or

18. Fixed Setup Cost & Backorder Cost
Fixed setup (or order) cost
First, set number of orders per year
Then, find annual fixed order costs
Backorder Cost
Inventory position is uniformly distributed between r+1 and r+Q
Averaging the backorder level over the range r+1 to r+Q

19. Stockout Cost in (Q,r) Model
Penalizes poor customer service
Charge a cost each time a stockout occurs
Charge a penalty that is proportional to the time a customer waits to have their order filled
Approximations
Type I (base stock) – computes # of stockouts per cycle, underestimates S(Q,r)
Type II – neglects B(Q,r) term, also underestimates S(Q,r)

20. Holding Costs in (Q,r) Model
Inventory holding cost = hI(Q,r)
First equation approximates and underestimates average inventory, since demand is variable and thus backorders sometimes occur
Exact formulation:

21. Backorder Cost Approach
Compute Q and r values that minimize:
approximate, since B(r) replaces B(Q,r)
Optimal Q* and r*
Then, assume normal distribution for lead time as in base stock model

22. Stockout Cost Approach
Compute Q and r values that minimize:
approximate, since B(r) replaces B(Q,r)
Optimal Q* and r*
Assume normal distribution for lead time
Note: larger Q results in smaller r* because a smaller reorder point is needed to achieve the same fill rate

23. (Q,r) Insights
Cycle stock increases as replenishment frequency decreases
Safety stock provides a buffer against stockouts
The base stock model is a (Q,r) model where Q=1
Increasing annual demand (D) increases Q
Increasing demand during leadtime increases r (thus, high annual demand or long leadtimes require inventory)
Increased demand variability increases r (for most parts where high fill rates are desired)
Increased holding costs (h) decreases both Q and r (if it is expensive to hold inventory, avoid doing so!)