EOQ Model(Decision problem)

The simple EOQ model attempts to balance holding and ordering costs Relevant Costs in EOQ
• Holding or Carrying cost (C)
• Ordering cost (P)
• Purchase cost of the item (V)

Where:
P = The ordering cost (dollars per order)
D = Annual demand or usage of the product (number of units)
C = Annual inventory carrying cost (as a percentage of product cost or value)
V = Average cost or value of one unit of inventory

An example
D = 4,800 Annual Demand
P = $40.00 Cost to Place an Order
V = $62.50 Value of one unit at Cost
C = 40% Annual Carrying Cost as a Percentage   

Optimization Model

• Objective function
    Minimize (Ordering costs + Inv. holding costs)
• Decision variable
    Order quantity Q
• Parameter
    Order cost (K) and Inv. holding cost (h)
• Constraint
    Positive order quantity

Assumptions of the Simple EOQ Model

• A continuous, constant, and known rate of demand.
• A constant and known replenishment cycle or lead time.
• A constant purchase price that is independent of the order quantity or time.
• A constant transportation cost that is independent of the order quantity or time.
• The satisfaction of all demand (no stockouts are permitted).
• No inventory in transit.
• Only one item in inventory, or at least no interaction among items.
• An infinite planning horizon.
• No limit on capital availability.

Dimensions of inventory models

• Deterministic versus Stochastic
• Indefinite versus finite planning horizon
• Independent versus dependent demand
• Single versus multiple:
    item
    location
    echelon (interrelated locations)
    indenture (interrelated items)

Lot Sizing Techniques

   
References for Part Period Balancing., LTC, LUC
• Materials Requirement Planning, http://www.shef.ac.uk
References for Wagner-Whitin :
• The Wagner-Whitin Model, Hierarchical Production Planning and Control, University of Massachusetts Amherst. PPT
• On the complexity of the economic lot-sizing problem with remanufacturing options, Wilco van den Heuvela,Econometric Institute and Erasmus Research, Institute of Management, Erasmus, University

Effects of Lot Sizing

• Lot-for-lot

+ ”Preserves” the MPS quantities
+ Suitable for JIT manufacturing
+ Generates smooth requirements for material and capacity
– No economic considerations

• Fixed order quantities

– Lumps together requirements to large orders
– Amplifies lumpiness through the BOM
– Fluctuating material and capacity requirements

• Variable quantity and cover-time

+ Economic considerations considering discrete requirements
- Estimation of cost parameters
– Covering many periods net requirements tends to create amplified variability of demand for material and capacity

Buffering Techniques

• Safety stock

    Physical safety quantity
    Used when quantity, demand or consumption is unreliable
    Net req. generated when safety stock is used (rec. by APICS)

• Safety lead time

    Safety in time, order receipt before requirement
    Used when lead times are stochastic
    Extends the lead time

• Hedging

    Safety in order quantities
    Used when yield is stochastic (e.g. scrap)
    Mainly used in master production scheduling

• Slack in the system (e.g. spare capacity)

Some Safety Stock Strategies

• Specified fill rate (demand filled from stock) or Specified service level (probability a stockout will not occur)
• Maximize $ demand filled from a given investment.
• Set SS based on specified number of Sigmas (Std.Dev., MAD, etc.)
• Set SS based on specified time supply.
• Minimize shortage occurrences for a given investment (# of orders with a problem.)
• Minimize transaction shortages for a given investment (# of problems in orders.)

Inventory Decision Rules

• Independent inventory Q,R policy

• What can happen: Q,R Policy

   
  Demand during lead time is larger than order size.
  If ordered only when replenishment comes, inventory is depleted.
  Pink line when backorders, black when demand is lost.

• Q, R Policy (amended with multiple R)

   
Examples: KanBan inventory in PSG or long lead-time processes, like replenishments from China.

• S,T Policy (S = 60 ,T = 7)

  
When lead-time is 5 days, and expected demand is 6 pcs/day, we order up to 60 + 5*6 = 90. As balance is 48, we order 42 pcs.

• Inventory decision rules

  Q is clear but S somewhat less clear. If we review inventory balance continuously, then when reorder point is reached and quantity Q ordered, it will lead to an expected opening inventory of S
  However, if review is periodic, then the inventory can be more or less below R, so S would be S = Q + (R-inventory).
  Policy Q,T is interesting. If demand during review T > Q, this policy does not really work. Unless we decide that it is still Q that is ordered but we can order or more times Q (n*Q).
  In practice, Q could be some physical logistics limit, like full truckload or a full pallet.
  In practice, S could be some periodic system, like shipping schedule or production cycle.
 

Parameter Calculation (Basic Model)

• Reorder Point Determination

a. Probability of not stocking out during lead time:
Safety stock = Z*δd
Where
– Z = value from the standard normal distribution
– δd = standard deviation of demand during replenishment lead time
Reorder Point = Z*δd + expected demand during lead time
Examples
– Z(1,645) = 95%
– Z(1,960) = 97,5 %
– Z(3,090) = 99,9 %
 
Minor Problems
• It calculates the probability of a stockout during replenishment lead time, not customer service level measured as fill-rate. They are not the same thing.
• What if delivery time is not certain but a variable, too?
• The formula applies only for normally distributed demand, not other demand distributions.
• How to incorporate demand forecasts?

b. Some Extensions: Variable lead time
If the LT distribution is binomial, then the joint distribution can be created manually.
• Bowersox gives a following approximate formula for calculating Z for variable lead time situation

joint δ = √(t* δd 2+ d2* δt2

Where
t = replenishment lead time
δt = replenishment lead time variance
d = demand during average replensihment lead time
δd = demand variance during average replensihment lead time

c. Other Distributions
• Normal distribution may sometimes be a reasonable assumption due  to central limit theorem.
• Normal distribution is symmetric.
• Physical inventory cannot be negative, so standard deviation must be clearly smaller than average (δd
   2< 0,5*X), which still gives 2,275% negative values for demand.
• Skewed distributions are likewise found in practice. Then incorrect normality assumption underestimates safety stock requirements.
• There are statistical tests for testing distributions.
• However, it can prove difficult to find analytical formulas for other distributions.
• One can always use empirical data and simulate its distribution and base decision on that distribution.

d. Consolidation benefits
• From probability theory you may recall that if X and Y are independent variables, then

Var (X + Y) = Var X + Var Y
• Now that we are talking about independent inventory management, this theorem gives a handy way to estimate consolidation benefits for safety stock.
• Likewise, it enables data conversions: if we have weekly standard deviation and lead time is two weeks, the theorem above can be used to calculate two-week standard deviation.

e. Parameter calculation with fill rate criterion
• Probability of a stockout ≠ customer fill rate
• The stockout magnitude must be weighted with its probability.
• In continuous distributions, this is achieved by a loss function (Vollman calls it service function).
• It is tabulated in Vollman and approximation formulas for spreadsheets do exist.
 
SL = 1 – δd*E(Z) /Q
E(Z) = (1 – SL)*Q/ δd
 
where
Q = order quantity
SL = Fill-rate (service level)
δd = standard deviation of demand (under repl. LT)
E(Z) = partial expectation of distribution (loss function)
Example
• If we use the values in previous examples, (Q=60, δd = 3*√5) we get for 95% fill rate
E(Z) = (1 – 0,95)*60/(3* √5)
• from E(Z) = 0,45 follows (through tabulated values) that Z ≈ - 0,05
• and safety stock is –0,05 * (3* √5) = - 0,34
• Reorder point is demand during lead time + safety
stock (6*5-0,34) which would become ROP = 29,66.

f. Parameter calculation with forecast
• Independent inventory safety stock is held against uncertainty. If forecasts are correct, no need for safety
stock. But they aren’t.
• Forecast quality is measured with mean absolute deviation (MAD).
• If you take the absolute values of normally distributed, its turns out that

E(|X|) = 0,8* δx
• So you can simply replace your demand variance δd with 1,25*MAD in the previous formulas
• Provided, of course, that your forecast error is ~ N.