graduate program in business information systems inventory decisions with uncertain factors aslı...

Graduate Program in Business Information Systems

Inventory Decisions with Uncertain Factors

Aslı Sencer

BIS 517- Aslı Sencer 2

Uncertainties in real life

Demand is usually uncertain.

Probability distributions are used to represent uncertain factors.

Ex: Demand is normally distributed OR

Demand is either 20,30,40 with respective probabilities 0.2, 0.5, 0.3.


Stochastic versus Deterministic Models Mathematical models involving probability are

referred to as stochastic models. Deterministic models are limited in scope

since they do not involve uncertain factors. But they are used to develop insight!

Stochastic models are based on “expected values”, i.e. the long run average of all possible outcomes!


Example: Drugstore

A drugstore stocks Fortunes.They sell each for $3 and unit cost is $2.10. Unsold copies are returned for $.70 credit. There are four levels of demand possible. How many copies of Fortune should be stocked in October?

Payoff Table:

Demand

Event Probability

ACTS

Q = 20 Q = 21 Q = 22 Q = 23D = 20 .2 $18.00 $16.60 $15.20 $13.80

D = 21 .4 8.00 18.90 17.50 16.10

D = 22 .3 8.00 18.90 19.80 18.40

D = 23 .1 8.00 18.90 19.80 20.70


Solution:

The expected payoffs are computed for each possible order quantity:

Q = 20 Q = 21 Q = 22 Q = 23$18.00 $18.44 $17.90 $16.79

Optimal stocking level, Q*=21 at an optimal expected profit of $18.44

If the probabilities were long-run frequencies, then doing so would maximize long-run profit.


Example: Drugstore Payoff Table(Figure 16-1)

123456789

101112131415161718

A B C D E F

PROBLEM: Fortune Magazine

Act 1 Act 2 Act 3 Act 4Events Probability Q = 20 Q = 21 Q = 22 Q = 23

1 D = 20 0.2 $18.00 $16.60 $15.20 $13.802 D = 21 0.4 $18.00 $18.90 $17.50 $16.103 D = 22 0.3 $18.00 $18.90 $19.80 $18.404 D = 23 0.1 $18.00 $18.90 $19.80 $20.70

Act 1 Act 2 Act 3 Act 4Q = 20 Q = 21 Q = 22 Q = 23

Expected Payoff $18.00 $18.44 $17.96 $16.79

PAYOFF TABLE EVALUATION

Problem Data

Act Summary

18C

=SUMPRODUCT($B$9:$B$12,C9:C12)


Single-Period Inventory Decision:The Newsvendor Problem Single period problem (periodic review) Demand is uncertain (stochastic) No fixed ordering cost Instead of h ($/$/period) we have hE ($/unit/period=ch) Instead of p ($/unit) we have pS and pR-c

Q: Order Quantity (decision variable)D: Demand Quantity

Costs:c = Unit procurement costhE = Additional cost of each item held at end of inventory cycle = unit inventory holding cost-salvage value to the supplierpS = Penalty for each item short (loss of customer goodwill)pR = Selling price


Modeling the Newsvendor Problem

The objective is to minimize total expected cost, which can be simplified as:

where is the expected demand.

QDifQ

QDifDSales

)()()( QBcppQBQchcQTEC RSE

QdifQdcppcQ

QdifdQhcQQTC

RS

E

)(


Optimal order quantity of the Newsboy Problem

Q* is the smallest possible demand such that

chcpp

cppQD

ERS

RS

]*Pr[


Example: Newsboy Problem

A newsvendor sells Wall Street Journals. She loses pS = $.02 in future profits each time a customer wants to buy a paper when out of stock. They sell for pR = $.23 and cost c = $.20. Unsold copies cost hE = $.01 to dispose. Demands between 45 and 55 are equally likely. How many should she stock?


Example: Solution

Discrete Uniform Distribution

Demand is either 45,46,47,..., 55 each with a probability of 1/11.

192

2001202302202302

......

...chcpp

cpp

ERS

RS

P(D<=Q*)=0.2 Q*=47 units.

12

Newsvendor Problem (Figure 16-3)

1234567

8

910111213141516171819202122232425262728293031

A B C D E F G

PROBLEM: Wall Street Journal

Parameter Values:Cost per Item Procured: c = 0.20Additional Cost for Each Leftover Item Held: hE = 0.01

Penalty for Each Item Short: pS = 0.02

Selling Price per Unit: pR = 0.23Number of demands for probability distribution = 11

Optimal Values:Optimal Order Quantity: Q* = 47Expected Demand: mu = 49.5Total Expected Cost: TEC(Q*) = $10.07Expected Shortages: B(Q*) = 2.66Probability of Shortage: P[D>Q*] = 0.80

Cumulative Number ofDemand Probability Probability shortages

45 0.05 0.05 0.046 0.06 0.11 0.047 0.09 0.20 0.048 0.12 0.32 1.049 0.17 0.49 2.050 0.20 0.69 3.051 0.12 0.81 4.052 0.08 0.89 5.053 0.06 0.95 6.054 0.04 0.99 7.055 0.01 1.00 8.0

SINGLE PERIOD INVENTORY MODEL -- NEWSVENDOR PROBLEM

BIS 517- Aslı Sencer


Multiperiod Inventory Policies When demand is uncertain, multiperiod inventory

might look like this over time.


Multiperiod Inventory Policies

The multiperiod decisions involve two variables: Order quantity Q Reorder point r

The following parameters apply: A = mean annual demand rate k = ordering cost c = unit procurement cost ps = cost of short item (no matter how long) h = annual holding cost per dollar value = mean lead-time demand


Multiperiod Inventory Policies: Discrete Lead-Time Demand

The following is used to compute the expected shortage per inventory cycle:

The following is used to compute the total annual expected cost:

rd

L dDrdrB Pr

rBQA

prQ

hckQA

Q,rTEC S

2


Multiperiod Inventory Policies: Discrete Lead-Time Demand Solution Algorithm.

Calculate the starting order quantity:

Determine the reorder point r*:

Determine optimal order quantity:

This procedure continues –using the last Q to obtain r and r to obtain the next Q- until no values change.

hcAk

Q2

1

Ap

hcQrD

S

1*Pr

hc

rBpkA*Q S 2


Example:

Annual demand for printer cartridges costing c = $1.50 is A = 1,500. Ordering cost is k = $5 and holding cost is $.12 per dollar per year. Shortage cost is pS = $.12, no matter how long. Lead-time demand

has the following distribution. Find the optimal inventory policy.


Example: Solution

The starting order quantity is:

r* = 7 cartridges. B(7) = (8–7)(.03) + (9–7)(.01) + (10–7)(.01)=.08 and

the optimal order quantity is:

28950112

5500121

..,

Q

93.

500,1)5.0(

2895.1)12.0(11*Pr

Ap

hcQrD

S

29050112

0850550012 ..

..,*Q


Example: Solution (cont’d.)

Q=290 leads to r=7, so the solution is optimal.The optimal inventory policy is:

r* = 7 Q* = 290

Optimal annual expected cost is:

71.52$08.0290

15005.047

2

290)5.1)(12.0(5

290

1500TEC(7,290)

20

Multiperiod Discrete BackorderingIteration 1

12345678910111213141516171819202122

2324252627282930313233

A B C D E F GMULTI-PERIOD EOQ MODEL (Backordering) - DISCRETE LEAD-TIME DEMAND

PROBLEM: Printer Cartridges

Parameter ValuesFixed Cost per Order: k = 5Annual Demand Rate: A = 1500Unit cost of Procuring an Item: c = 1.5Annual Holding Cost per Dollar Value: h = 0.12Shortage Cost per Unit: pS = 0.5Number of demands for probability distribution = 11

Optimal Values:Optimal Order Quantity: Q* = 288.68Optimal Reorder Point: r* = 7Expected Lead-Time Demand: mu = 4Total Expected Cost: TEC(Q*) = 52.7094$ Expected Shortage: B(r*) = 0.08Probability of Shortage: P[D>r*] = 0.05

Cumulative Number ofDemand Probability Probability Shortages

0 0.01 0.01 01 0.07 0.08 02 0.16 0.24 03 0.20 0.44 04 0.19 0.63 05 0.16 0.79 06 0.10 0.89 07 0.06 0.95 08 0.03 0.98 19 0.01 0.99 210 0.01 1.00 3

14

1516

171819

G=SQRT((2*G7*G6)/(G9*G8))

=INDEX(C23:C42,MATCH(LOOKUP(1-((G9*G8*G14)/(G10*G7)),E23:E42,C23:C42),C23:C42,0)+1)=SUMPRODUCT(C23:C42,D23:D42)

=(G7/G14)*G6+G9*G8*(G14/2+G15-G16)+G10*(G7/G14)*G18=SUMPRODUCT(D23:D42,F23:F42)=1-VLOOKUP(G15,C23:E42,3)


21

123456789

10

111213141516171819202122

2324252627282930313233



Parameter ValuesFixed Cost per Order: k = 5Annual Demand Rate: A = 1500Unit cost of Procuring an Item: c = 1.5Annual Holding Cost per Dollar Value: h = 0.12Shortage Cost per Unit: pS = 0.5

Number of demands for probability distribution = 11

Optimal Values:Optimal Order Quantity: Q* = 290Optimal Reorder Point: r* = 7Expected Lead-Time Demand: mu = 4Total Expected Cost: TEC(Q*) = 52.71$ Expected Shortage: B(r*) = 0.08Probability of Shortage: P[D>r*] = 0.05

Cumulative Number ofDemand Probability Probability Shortages

0 0.01 0.01 01 0.07 0.08 02 0.16 0.24 03 0.20 0.44 04 0.19 0.63 05 0.16 0.79 06 0.10 0.89 07 0.06 0.95 08 0.03 0.98 19 0.01 0.99 2

10 0.01 1.00 3

14

1516

171819

G=SQRT((2*G7*(G6+'9'!G18*G10))/(G9*G8))

=INDEX(C23:C42,MATCH(LOOKUP(1-((G9*G8*G14)/(G10*G7)),E23:E42,C23:C42),C23:C42,0)+1)=SUMPRODUCT(C23:C42,D23:D42)

=(G7/G14)*G6+G9*G8*(G14/2+G15-G16)+G10*(G7/G14)*G18=SUMPRODUCT(D23:D42,F23:F42)=1-VLOOKUP(G15,C23:E42,3)

Multiperiod Discrete BackorderingIteration 10

14G

=SQRT((2*G7*(G6+'9'!G18*G10))/(G9*G8))


22

12345678910111213

14151617181920212223



Parameter ValuesFixed Cost per Order: k = 5Annual Demand Rate: A = 1500Unit cost of Procuring an Item: c = 1.5Annual Holding Cost per Dollar Value: h = 0.12Shortage Cost per Unit: pS = 0.5Number of demands for probability distribution = 11

Iteration, i Qi ri B(ri) TEC(Qi,ri)

1 289 7 0.08 52.71$ 2 290 7 0.08 52.71$ 3 290 7 0.08 52.71$ 4 290 7 0.08 52.71$ 5 290 7 0.08 52.71$ 6 290 7 0.08 52.71$ 7 290 7 0.08 52.71$ 8 290 7 0.08 52.71$ 9 290 7 0.08 52.71$ 10 290 7 0.08 52.71$

Multiperiod Discrete BackorderingSummary


graduate program in business information systems inventory decisions with uncertain factors aslı...

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