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Inventory Control



1. EOQ Model (Decision problem 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)


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)



References for Part Period Balancing., LTC, LUC

References for Wagner-Whitin :

  • 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

  • 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)

  • 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.)

  • 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.

  • Reorder Point Determination

  1. Probability of not stocking out during lead time:

Safety stock = Z*δd


– 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


– 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?

  1. 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


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

  1. 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.

  1. 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.

  • 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

= order quantity
SL = Fill-rate (service level)
δd = standard deviation of demand (under repl. LT)
E(Z) = partial expectation of distribution (loss function)


  • 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.

  1. 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.

  • Juha-Matti Lehtonen, Dr. Tech, .TU-22.176 OPERATIONS MANAGEMENT, Helsinki University of Technology,  Industrial Engineering and Management

  • TPPE37 Manufacturing Control, Department of Production Economics (IPE), Linköping Institute of Technology