isye3104chapter2-inventory+control+Q%2Cr
-
Upload
lieng723438032 -
Category
Documents
-
view
13 -
download
0
Transcript of isye3104chapter2-inventory+control+Q%2Cr
ISyE 3104 Fall 2014 © Georgia Tech, 2014 1
Inventory Control PART 3: STOCHASTIC DEMAND: Q ,r MODEL
ISyE 3104 Fall 2014 © Georgia Tech, 2014 2
The Single Product (Q,r) Model
Consider a central distribution facility which orders from a manufacturer and delivers to retailers. The distributor periodically places orders to replenish its inventory:
Random demand
Fixed lead time
Fixed setup/order cost
Why hold inventory?
ISyE 3104 Fall 2014 © Georgia Tech, 2014 3
The Single Product (Q,r) Model
Motivation: Either
1. Fixed cost associated with replenishment orders and cost per backorder.
2. Constraint on number of replenishment orders per year and service constraint.
Objective: Under (1) costbackorder cost holdingcost setup fixed min
Q,r
As in EOQ, this makes
batch production attractive.
The relevant costs are: Set-up each time an order is placed Holding at per unit held per unit time ( i. e., per year) Penalty per unit of unsatisfied/backordered demand
ISyE 3104 Fall 2014 © Georgia Tech, 2014 4
Inventory Profile of (Q, r) Policy
Time
Inve
nto
ry L
evel
Q+r
r
0
Lead Time Lead Time
Inventory Position
Q Q
On-hand Inventory
Q
s
ISyE 3104 Fall 2014 © Georgia Tech, 2014 5
The (Q, r) Policy
Policy: If the inventory level is at or below r, order Q
r is the reorder point, and Q is the order quantity ◦ r is chosen to protect against uncertainty of demand during the lead time
◦ Q is chosen to balance the holding and set-up costs
ISyE 3104 Fall 2014 © Georgia Tech, 2014 6
Summary of (Q,r) Model Assumptions
1. Demand is uncertain, but stationary over time and distribution is known.
2. Continuous review of inventory level.
3. Fixed replenishment lead time l
4. Constant replenishment batch sizes (Q)
5. Stockouts are backordered.
ISyE 3104 Fall 2014 © Georgia Tech, 2014 7
(Q,r) Model Derivation
ISyE 3104 Fall 2014 © Georgia Tech, 2014 8
(Q,r) Notation
cost backorder unit annual
stockout per cost
cost holding unit annual
item an of cost unit
order per cost fixed
time lead during demand of cdf ) ()(
time lead during demand of pmf
time lead entreplenishm during demand of deviation standard
time lead entreplenishm during demand expected ][
time lead entreplenishm during demand (random)
constant) (assumed time lead entreplenishm
year per demand expected
b
k
h
c
A
xXPxG
xXPg(x)
XE
X
D
)(
ISyE 3104 Fall 2014 © Georgia Tech, 2014 9
(Q,r) Notation (cont.)
levelinventory average),(
levelbackorder average ),(
rate) (fill level service average ),(
frequencyorder average )(
by impliedstock safety
pointreorder
quantityorder
rQI
rQB
rQS
QF
rrs
r
Q
Decision Variables:
Performance Measures:
ISyE 3104 Fall 2014 © Georgia Tech, 2014 10
Costs in (Q,r) Model with Backordered Demand
),(),(),( rQhIrQbBAQ
DrQY
ISyE 3104 Fall 2014 © Georgia Tech, 2014 11
Inventory Profile of (Q, r) Policy
Time
Inve
nto
ry L
evel
Q+r
r
0
Lead Time Lead Time
Inventory Position
Q Q
On-hand Inventory
Q
s
Expected Inventory Position
ISyE 3104 Fall 2014 © Georgia Tech, 2014 12
Inventory Costs in (Q,r) Model
Recall: Inventory Position = On hand Inventory + On order – Backorders - Committed
On average: expected inventory position declines from Q +r to r+1
(Q+ r)+ (r+1)
2= I(Q, r)+q -B(Q, r)
I(Q, r) =Q+1
2+ r -q +B(Q, r)
ISyE 3104 Fall 2014 © Georgia Tech, 2014 13
Backorder Costs
Backorder cost = bB(Q,r)
)(),(
)]()([1
),(
)(1
),(
rBrQB
QrBrBQ
rQB
dxxBQ
rQB
Qr
r
Averaging the backorders over all ranges of Q
Could be approximated by
ISyE 3104 Fall 2014 © Georgia Tech, 2014 14
Backorder Approximation
/)(),()](1)[()(
)()1()(1
rzwherezzrrB
dxxgrxrBr
:to simplified is this then ddistribute normally is demand If
)](1)[()())(1()()()(1
0
rGrrgxGxgrxrBr
xrx
simpler version for
spreadsheet
computing, but
only works for
Poisson demand
Recall from Base Stock model: If g(x) is continuous:
If g(x) is discrete:
ISyE 3104 Fall 2014 © Georgia Tech, 2014 15
(Q,r) Model with Backorder Cost
Objective Function:
),(),(),( rQhIrQbBAQ
DrQY
))(2
1()(),(
~),( rBr
QhrbBA
Q
DrQYrQY
ISyE 3104 Fall 2014 © Georgia Tech, 2014 16
Finding Q* and r*
ISyE 3104 Fall 2014 © Georgia Tech, 2014 17
Two Solution Procedures to find Q* and r*:
Based on minimizing the expected costs
Based on achieving a predetermined service level: Type 1 service: not stocking out in a cycle
Type 2 service: proportion of demand met from stock
ISyE 3104 Fall 2014 © Georgia Tech, 2014 18
Solution Approach 1: Minimizing expected costs
ISyE 3104 Fall 2014 © Georgia Tech, 2014 19
Results of Approximate Optimization
Assumptions: ◦ Q,r can be treated as continuous variables
◦ G(x) is a continuous cdf
Results:
zrbh
brG
h
ADQ
**)(
2*
if G is normal(,),
where (z)=b/(h+b)
Note: this is just the EOQ formula
Note: this is just the
base stock formula
ISyE 3104 Fall 2014 © Georgia Tech, 2014 20
Service Level Approximations
Type I (base stock):
Type II:
)(),( rGrQS
Q
rBrQS
)(1),(
Note: computes number
of stockouts per cycle,
underestimates S(Q,r)
Note: neglects B(r,Q)
term, underestimates S(Q,r)
ISyE 3104 Fall 2014 © Georgia Tech, 2014 21
(Q,r) Example
Stocking Repair Parts:
D = 14 units per year
c = $150 per unit
h = 0.1 × 150 + 10 = $25 per unit
l = 45 days
ϴ = (14 × 45)/365 = 1.726 units during replenishment lead time
A = $10
b = $40
Demand during lead time is Poisson
ISyE 3104 Fall 2014 © Georgia Tech, 2014 22
Values for Poisson(q) Distribution
r g(r) G(r) B(r)
0 0.178 0.178 1.726
1 0.307 0.485 0.904
2 0.265 0.750 0.389
3 0.153 0.903 0.140
4 0.066 0.969 0.042
5 0.023 0.991 0.011
6 0.007 0.998 0.003
7 0.002 1.000 0.001
8 0.000 1.000 0.000
9 0.000 1.000 0.000
10 0.000 1.000 0.000
)](1)[()( rGrrg
ISyE 3104 Fall 2014 © Georgia Tech, 2014 23
Calculations for Example
615.04025
40
43.415
)14)(10(22*
bh
b
h
ADQ
From the table r* = 2, Or, using the normal approximation:
2107.2)314.1(29.0726.1*
29.0615.0)29.0(
zr
z so ,
ISyE 3104 Fall 2014 © Georgia Tech, 2014 24
Performance Measures for Example
823.2049.0726.122
14*)*,(*
2
1**)*,(
049.0]003.0011.0042.0140.0[4
1
)]6()5()4()3([1
)(*
1*)*,(
904.0]003.0389.0[4
11
)]42()2([1
1*)]*(*)([*
11**
5.34
14
**)(
**
1*
rQBrQ
rQI
BBBBQ
xBQ
rQB
BBQ
QrBrBQ
),rS(Q
Q
DQF
Qr
rx
ISyE 3104 Fall 2014 © Georgia Tech, 2014 25
Observations on Example
•Orders placed at rate of 3.5 per year
•Fill rate fairly high (90.4%)
•Very few outstanding backorders (0.049 on average)
•Average on-hand inventory just below 3 (2.823)
ISyE 3104 Fall 2014 © Georgia Tech, 2014 26
Varying the Example
Change: suppose we order twice as often so F=7 per year, then Q=2 and:
which may be too low, so increase r from 2 to 3:
This is better. For this policy (Q=2, r=4) we can compute B(2,3)=0.026, I(Q,r)=2.80.
Conclusion: this has higher service and lower inventory than the original policy (Q=4, r=2). But the cost of achieving this is an extra 3.5 replenishment orders per year.
826.0]042.0389.0[2
11)]()([
11),( QrBrB
QrQS
936.0]011.0140.0[2
11)]()([
11),( QrBrB
QrQS
ISyE 3104 Fall 2014 © Georgia Tech, 2014 27
Solution Approach 2: Meeting Predetermined Service Levels
ISyE 3104 Fall 2014 © Georgia Tech, 2014 28
Finding Q* and r* based on Service Levels in (Q,r) Systems
In many circumstances, the backorder costs (b) are difficult to estimate.
For this reason, it is common business practice to set inventory levels to meet a specified service objective instead. The two most common service objectives are:
1) Type 1 service: Choose r* so that the probability of not stocking out in the lead time is equal to a specified value.
2) Type 2 service (fill rate). Choose both Q* and r* so that the proportion of demands satisfied from stock equals a specified value.
ISyE 3104 Fall 2014 © Georgia Tech, 2014 29
Service Level Approximations
Type I:
Type II:
)(),( rGrQS
Q
rBrQS
)(1),(
ISyE 3104 Fall 2014 © Georgia Tech, 2014 30
Finding Q* and r* based on Service Levels in (Q,r) Systems
For type 1 service, if the desired service level is α then one finds r* from G(r*)= α and Q*=EOQ.
For type 2 service, if the desired service level is β then ◦ set Q*=EOQ
◦ find r* to satisfy
*)(*
11 rB
Q
ISyE 3104 Fall 2014 © Georgia Tech, 2014 31
(Q,r) Example
Stocking Repair Parts:
D = 14 units per year
c = $150 per unit
h = 0.1 × 150 + 10 = $25 per unit
l = 45 days
ϴ = (14 × 45)/365 = 1.726 units during replenishment lead time
A = $10
Demand during lead time is Poisson
ISyE 3104 Fall 2014 © Georgia Tech, 2014 32
Values for Poisson(q) Distribution
r p(r) G(r) B(r)
0 0.178 0.178 1.726
1 0.307 0.485 0.904
2 0.265 0.750 0.389
3 0.153 0.903 0.140
4 0.066 0.969 0.042
5 0.023 0.991 0.011
6 0.007 0.998 0.003
7 0.002 1.000 0.001
8 0.000 1.000 0.000
9 0.000 1.000 0.000
10 0.000 1.000 0.000
ISyE 3104 Fall 2014 © Georgia Tech, 2014 33
Calculations for Example
43.415
)14)(10(22*
h
ADQ
If the desired type 1 service level is 90% => G(r*) = 0.9 =>
341.3)314.1(28.1726.1*
28.19.0)(
zr
zz so ,
If the desired type 2 service level is 90%
4.0*)(9.0*)(*
11 rBrB
Q
2*
)()](1)[()(
r
zzrrB
using
ISyE 3104 Fall 2014 © Georgia Tech, 2014 34
Performance Measures for Example
*)*,(*2
1**)*,(
)(*
1*)*,(
*)]*(*)([*
11**
**)(
**
1*
rQBrQ
rQI
xBQ
rQB
QrBrBQ
),rS(Q
Q
DQF
Qr
rx
ISyE 3104 Fall 2014 © Georgia Tech, 2014 35
Single Product (Q,r) Insights
Increasing D tends to increase optimal order quantity Q.
Increasing ϴ increase the reorder point.
Increasing the variability of the demand process tends to increase the optimal reorder point (provided z > 0).
Increasing the holding cost tends to decrease the optimal order quantity and reorder point.