Lot size/Reorder level (Q,R) Models

Lot size/Reorder level (Q,R) Models

ISYE 3104 ? Fall 2013

? Esma Gel, Pinar Keskinocak, 2013

Recap: Basic EOQ

Inventory I(t)

Q -d

R=d

T Q d

T

2T

3T

4T

time

Lead time

Place an order when the inventory level is R. The order arrives after time periods Q was the only decision variable R could be computed easily because D was deterministic

1

Uncertain demand

Both Q and R are decision variables Cycle time is no longer constant!

Inventory

R

s2

s1

s3

Q

T1

Q Q

time

T2

T3

(Q,R) Decisions

We choose R to meet the demand during lead time

Service levels: Protect against uncertainties in demand (or lead time)

Balance the costs: stock-outs and inventory

Tradeoff in Q: Fixed cost versus holding cost

Objective: Minimize

fixed cost + holding cost + stockout (backorder) cost

2

Demand during lead time

Inventory

R

Stockout

Reorder level

What will happen if demand follows one of these patterns?

Excess inventory

Time

Demand during lead time

Inventory

Often the probability distribution of demand during lead time follows a Normal pattern

R

P(D>R)=

Probability

of stockout

Time

Expected demand R

during lead time

3

(Q,R) Model Assumptions

Continuous review Demand is random and stationary. Expected demand is d per

unit time. Lead time is Costs

K: Setup cost per order h: Holding cost per unit per unit time c: Purchase price (cost) per unit p: Stockout (backorder) cost per unit Demand during lead time is a continuous random variable D with pdf (density function) f(x) and cdf (distribution function) F(x) Mean= and standard deviation=

(Q,R) Model ? Expected total cost per unit time

Holdingcost Fixed cost Shortage cost

C(Q) h s Q K p n(R)

2 T

T

Recap : T Q d

s Average inventory level before an order arrives

(Reorder level) (expected demand during leadtime) R-

n(R) Expected shortage per cycle

D R shortage D-R

D R shortage 0

Standard loss function

R

n(R) 0 f (x)dx (x R) f (x)dx (x R) f (x)dx L(z)

0

R

R

4

(Q,R) Model ? Expected total cost per unit time

Holdingcost Fixed cost Shortage cost

C(Q) h s Q K p n(R)

2 T

T

Recap : T Q d

s Average inventory level before an order arrives

(Reorder level) (expected demand during leadtime) R-

n(R) Expected shortage per cycle

D R shortage D-R

D R shortage 0

Standard loss function

R

n(R) 0 f (x)dx (x R) f (x)dx (x R) f (x)dx L(z)

0

R

R

(Q,R) Model ? Expected total cost per unit time

Holdingcost Fixed cost Shortage cost

C(Q) h s Q K p n(R)

Recap : T Q

2 T

TSame expression asd

s Average inventory level btehfoere"eaxnpoerdceter darrives

(Reorder level) (expectneudmdebmearnodfduring leadtime) R-

n(R) Expected shortage per cysctloeckouts" in the

newsvendor model

D R shortage D-R

(Q replaced by R)

D R shortage 0

Standard loss function

R

n(R) 0 f (x)dx (x R) f (x)dx (x R) f (x)dx L(z)

0

R

R

5

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