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