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Stochastic Models Stochastic Models
InventoryControl
Inventory FormsInventory Forms
Form Raw Materials Work-in-Process Finished Goods
Inventory FunctionInventory Function
Safety Stock inventory held to offset the risk of unplanned
demand or production stoppages Decoupling Inventory
buffer inventory required between adjacent processes with differing production rates
Synchronized Production In-transit (pipeline)
materials moving forward through the value chain order but not yet produced/received
Inventory FunctionInventory Function
Cycle Inventory orders in lot size not equal to demand
requirements to lower per unit purchase costs
Decoupling Inventorybuffer inventory required between adjacent processes
with differing production rates Synchronized Production
In-transit (pipeline)materials moving forward through the value chainorder but not yet produced/received
Inventory FunctionInventory Function
Seasonal Inventory produce in low demand periods to meet the needs
in high demand periods anticipatory - produce ahead of planned downtime
In-transit (pipeline) materials moving forward through the value chain order but not yet produced/received
Cycle Inventory orders in lot size not equal to demand requirements to lower per unit purchase costs
Inventory CostsInventory Costs
Item Cost (C) Order Cost (S)
Process Setup Costs Holding Costs (H)
Function of time in inventory, average inventory level, material handling, utilities, overhead, ...
Often calculated as a % rate of inventory cost (iC)
Stockout or Shortage Costs (s) reflects costs associated with lost opportunity
Economic Order Economic Order QuantityQuantity
Q
rtime
LT
1
D
order arrives
Q = reorder quantity r = reorder pointD = demand rate L = leadtime T = inventory cycle
Economic Order Economic Order QuantityQuantity
Q = reorder quantity r = reorder pointD = demand rate L = leadtime T = inventory cycle
Q
rtime
LT
Avg. Inventory
2
Inventory CostInventory Cost
TRC = Total Relevant Cost= Order Cost + Holding Costs
= cost per cycle
TCU = Total Relevant Cost per Unit Time= TRC/T
S HQT1
2ST HQ1
Cost per unit TimeCost per unit Time
But, T = length of a cycle
QD
TCUSD
QHQ 1
2
Cost per Unit TimeCost per Unit Time
Total Cost per Unit Time
0
5
10
15
20
25
30
500 1,000 1,500 2,000
Order Q
Co
st
Order
Hold
TCU
Cost per Unit TimeCost per Unit Time
Total Cost per Unit Time
0
5
10
15
20
25
30
500 1,000 1,500 2,000
Order Q
Co
st
Order
Hold
TCU
Note: that minimal per unit cost occurs whenholding cost = order cost (per unit time)
Economic Order Economic Order QuantityQuantity
Find min TCU
TCU
Q QSDQ HQ 0 1
2( )
-SDQ0 2 12
H
Q*SD
iC
SD
H
2 2
ExampleExample
The monthly demand for a product is 50 units. The cost of each unit is $500 and the holding cost per month is estimated at 10% of cost. It costs $50 for each order made. Compute the EOQ.
Q**50*50
.1*500
2= 10
Sol:Sol:
Optimal Inventory Optimal Inventory CostCost
Recall TCUSD
QHQ 1
2
TCU*SD
Q*HQ* 1
2
TCU*SD
SD
H
2H1
2
SD
H
2
Optimal Inventory Optimal Inventory CostCost
TCU HSD iCSD* 2 2
Example:Example:
2*.1*500*50*50TCU *
= $500 per month
Orders per yearOrders per year
HD
ND
Q S * 2
N = number of orders per year
Example:Example: D = 50 / month, Q* = 10
D = 50 x 12 = 600 / yr. H = .1x12x500 = $600 / unit-yr
N 600*600
2*50= 60
Cycle TimeCycle Time
TQ
D
S
HD
* 2
T = cycle time
Example:Example: D = 50 units/month, Q* = 10
T2*50
50*5010
50= .2 months = 6 days
Reorder PointReorder Point
L= lead time r = reorder point
= inventory depleted in time L= L*D
Example: Example: Lead time for company is 2 days. Demand is 50 units per month or 1.67 units/day.
r= 2*1.67 = 3.33Reorder at 4 units
Lead TimeLead Time
Example 2:Example 2: Suppose our lead time is closer to 8 days.
r = 8*1.67 = 13.33
but, recall we only order 10 units at a time
r = 13.33 - 10 = 3.33
Example (cont.)Example (cont.)
Reorder at 4 units 1 cycle ahead.
10
4time
L
T
order arrivesreorder
SensitivitySensitivity
Q*SD
H
2Recall that
Now suppose we deviate by p amount so that Q = Q*(1+p). What affect does this have on total cost? Let
PCP = Percentage Cost Penalty
PCPTCU Q TCU Q
TCU Qx
( ) ( )
( )
*
* 100
Senstivity (cont.)Senstivity (cont.)
TCU QSD
Q pHQ p( )
( )( )*
*
1
12 1
TCUSD
QHQ 1
2Recall
Miracle 1 Occurs
2TCU Q SDH pp
( ) ( )
1
21
1
1
Sensitivity (cont.)Sensitivity (cont.)Recall TCU HSD* 2
2SDH pp
( )
1
21
1
1PCP =
HSD2
HSD2x 100
= 50 pp
( )
1
1
1100
Sensitivity (cont.)Sensitivity (cont.)
PCP = 50 pp
( )
1
1
1100
Miracle 2 Occurs
PCPp
p
50
1
2
ExampleExample
Recall that Q* = 10. Suppose now that a minimum order of 15 is introduced. Compute the percentage cost penalty (increase).
p
15 10
105.
PCP
50 5
1 58 3
2(. )
..
Total relevant costs increase 8.3%
Example 2Example 2
Suppose demand forecast increases by 25% so that D = 50(1.25) = 62.5. Then
TCU * * . * *2 62 5 50 50 559
or TCU* increases by 11.8%
ShortagesShortages
Im
rtime
LT T1 T2
Q
Q-R
T1 = time inventory carried H = holding costT2 = time of stockout S = order cost Im = max inventory level p = cost per unit short per unit time
Inventory CostsInventory Costs
TCR = order + holding + shortages
S HImT p Q Im T12
121 2
( )
Miracle 3 Occurs
HDS
QH
p
p* 2
H
DSR
H
p
p* 2
+
ExampleExample
Suppose we allow backorders for our previous example. We estimate that the cost of a backorder is $1 per unit per day ($30 / month). Then
*50*50
Q50
* 5030
30
2
= 16.3 = 17 units
Production Model Production Model (ELS)(ELS)
Im
timeT1
1
P-D
T2 T
Q = batch size order quantity D = demand rate P = production rate P-D = replenish rate during T1
S = setup costs H = holding cost /unit-time Im = max inventory level
Production Model Production Model (ELS)(ELS)
Im
timeT1
1
P-D
T2 T
T = cycle length = T1 + T2 = Q/D T1 = length of production run = Q/P T2 = depletion time = Im/D Im = max inventory level
= (P-D)T1 = (P-D)Q/P
CostsCosts TC = total costs per cycle = order + holding
SH P D QT
P
1
2
( )
TCU = Cost per unit time TC T /
S
T
H P D Q
P
( )1
2
SD
Q
H P D Q
P
( )
2
Optimal Q* (ELS)Optimal Q* (ELS)
TCU
Q
SD
Q
H P D
P
0
22
( )
Solving for Q,
QSDP
H P D
EOQDP
*
( )
2
1
Max InventoryMax Inventory
Im = max inventory
= (P-D)T1
= (P-D)Q*/P
I QD
PELS
D
Pm
* 1 1
SummarySummary
Im
timeT1
1
P-D
T2 T
QSDP
H P D
EOQ*
( )
2DP1
I QD
Pm
* 1 DP1 EOQ
Probabilistic ModelsProbabilistic Models
Im
time
L
R=B+LDB
Q*
D = demand rate B= buffer stock R= reorder point = B+LD DL = actual demand from time of order to
time of arrival
Probabilistic ModelProbabilistic Model
Let = max risk level for out of stock condition
Idea: we want to set a buffer level B so that the probability of running out of stock is < .
> P{out of stock}= P{demand in DL > R}
= P{DL > B+LD}
ExampleExample
Prob: Suppose S=$100, H=$.02/day, L=2 days.
D = daily demand N(100, 10).
From EOQ model, Q* = 1,000 units
Find: Buffer level, B, such that probability of out of stock < .05.
SolutionSolution
DL = demand for 2 days = D1 + D2
Question: D1 & D2 are identically independently distributed normal variates with mean and standard deviation =10.
What can we say about the distribution of DL?
Prob. ReviewProb. Review
Suppose we have a random variable XL given by
X = Y1 + Y2
Then
E[X] = E[Y1] + E[Y2]
If Y1 & Y2 are independent, then
x 212
22
1 22 cov( , )y y
x 212
22
200
Solution (cont.)Solution (cont.)
Recall DL = demand for 2 days = D1 + D2
~ N(L,L)
Then L = E[DL] = E[D1] + E[D2] = 200
L D D2 2 2 2 2
1 210 10 200
L 14 14.
Solution (cont.)Solution (cont.)
DL ~ N(200, 14.14)
< P{DL > B + LD}
= P{DL > B + L}
PD Bl l
L L
P ZB
L
Solution (cont.)Solution (cont.)
Recall for our problem that =.05 and L=14.14.
Then, .
.05
14 14
P ZB
0Z=1.645
=.05B
14 141 645
..
B = 23.3
SummarySummary
For D ~ N(100,10), L = 2 days, Q* = 1,000 units, and = risk level = .05
DL ~ N(L=200, L=14.14)
B = ZL = 23.3
R = B + DL = B + L = 23.3 + 200 = 224
Optional Optional ReplacementReplacement
In the continuous review model, an order of Q* is made whenever inventory level reaches the reorder point R.
We can also utilize periodic review systems with variable order quantities. The two most common are
Optional Replacement (s,S)P system
Im
time
L
R=B+LDB
Optional Optional ReplacementReplacement
At t=1, inventory level is above minimum stock level s, no order is made.
At t =2, inventory level is below s, order up to S
s = R = B+DLS = Q*
S
time
s
1 2 3 4 5
P SystemP System
Order up to Target level T at each review interval P.
Let DP+L = demand in review period + lead time
P+L = standard deviation of demand in period P+L
= level of risk associated with a stockout
T = DP+L + ZP+L
T
time1 2 3 4 5
Newsboy Problem Newsboy Problem
Often inventory for a single product is met only once; e.g. News Stand (can’t sell day old papers)
Pet Rocks Christmas Trees
If Q > D, incur costs for Q but revenue only for D
If Q < D, incur opportunity costs in form of lost sales
Newsboy (cont.)Newsboy (cont.)
Objective: Determine best order quantity which maximizes expected profit
Payoff Matrix:
Rij = payoff for order quantity Qi and demand level Dj
P = profit per unit sold L = loss per unit not sold
RPQ if Q D
PD L Q D if Q Dij
i i j
i i j i j
( )
Newsboy (cont.)Newsboy (cont.)
Expected Payoff:
EP Q P Ri Dj
m
ijj( )
1
where
EP(Qi) = expected payoff for order quantity Qi
P = probability of demand level j
Rij = payoff for order quantity Qi and demand level Dj
Dj
Example; Example; NewsboyNewsboy
Boy Scout troop 53 plans to sell Christmas trees to earn money. Each tree costs the troop $10 and can be sold for $25. They place no value on lost sales due to lack of trees, L=0. Demand schedule is shown below.
Demand P{demand}100 0.10120 0.15140 0.25160 0.25180 0.15200 0.10
Example (cont.)Example (cont.)
Payoff Matrix
P = Profit = $25 - $10 per tree soldL = Loss = $-10 per tree not sold
Demand LevelOrder Q 100 120 140 160 180 200
100 1,500 1,500 1,500 1,500 1,500 1,500120 1,300 1,800 1,800 1,800 1,800 1,800140 1,100 1,600 2,100 2,100 2,100 2,100160 900 1,400 1,900 2,400 2,400 2,400180 700 1,200 1,700 2,200 2,700 2,700200 500 1,000 1,500 2,000 2,500 3,000
Example (cont.)Example (cont.)
Expected Payoff:
Demand Level0.1 0.15 0.25 0.25 0.15 0.1 Expected
Order Q 100 120 140 160 180 200 Payoff100 150 225 375 375 225 150 1,500
120 130 270 450 450 270 180 1,750
140 110 240 525 525 315 210 1,925
160 90 210 475 600 360 240 1,975
180 70 180 425 550 405 270 1,900
200 50 150 375 500 375 300 1,750
Order quantity has largest expected payoff of $1,975
order 160 trees