IE 431
Production Planning II
Inventory Theory II.
Dynamic Lot Sizing Models
1. Lot: The amount of a (by)product which is produced/purchased in one shot.
2. Silver-Meal Method (SM). It generalizes many simple rules as Fixed Period
Demand (incl. Lot for Lot (L4L))
Assumptions:
(i)
The demand is at the beginning of the period.
(ii)
The inventory is at the end of the period.
(iii)
The demand is known for n coming period: D1, D2,…, Dn,
(iv)
Holding cost and order cost are constants.
Method: The average cost K(m) is determined if the order is made for m
periods, where 1  m  n. The smallest m* is selected such that the average
cost is higher for m*+1 than for m*.
Formulas:
K(1)=A, K(2)=(A+hD2)/2, K(3)=(A+ hD2+2hD3)/3
K(m)=( A+ hD2+2hD3+…+(m-1)Dm)/m
m*=min { m  K(m) < K(m+1) }
Q= D1+D2+…+Dm
Example: A=50, h=0.5, D=(100,100,50,50,210)
3. Least Unit Cost (LUC). It has the same assumptions as the Silver-Meal heuristics
but the decision is made on the average cost per unit.
Formulas:
K(1)=A/D1, K(2)=(A+hD2)/(D1+D2), K(3)=(A+ hD2+2hD3)/ (D1+D2+D3)
K(m)=( A+ hD2+2hD3+…+(m-1)Dm)/(D1+D2+…+Dm)
m*=min { m  K(m) < K(m+1) }
Q= D1+D2+…+Dm
Example: A=50, h=0.5, D=(100,100,50,50,210)
4. Part Period Balancing (PPB).
Observation: At EOQ, i.e. at the Harris formula, the order cost equals to the
inventory holding cost (above the safety stock).
Assumptions. As before.
A new unit of storing. 1 part period = the storing of 1 unit in 1 time period.
Formulas.
PPm=the number of part periods if the order is made for the next m periods.
PP1=0
PP2=D2
PP3=D2+2D3
…
PPm= D2+2D3+…+ (m-1)Dm
Method: Select the time period such that the inventory holding cost is
approximately equal to the order cost, i.e.
A  h PPm.
Example: A=50, h=0.5, D=(100,100,50,50,210).
5. Wagner-Whitin Model.
It determines optimal order policy for finite many time periods.
Key observation: In an optimal solution a time period is either inherits a
positive inventory or has production/order but not both.
No formulas are given.
6. Peterson-Silver Rule. When shall the EOQ formula be applied?
Generally: When the demand is uniform.
The measure of uniformness:
Variability coefficient = V = (variance of demand per period)/
(square of average demand per period)
The rule. If V<0.25 the apply EOQ, otherwise something else.