Deterministic Inventory Management

Transcript Deterministic Inventory Management

INVENTORY MANAGEMENT

Operations Management Dr. Ron Lembke

Purposes of Inventory

     Meet anticipated demand  Demand variability  Supply variability Decouple production & distribution  permits constant production quantities Take advantage of quantity discounts Hedge against price increases Protect against shortages

2006 2007 13.81

1857 24.0% 446 801 58 1305 9.9

US Inventory, GDP ($B)

14,000 12,000 10,000 8,000 6,000 4,000 2,000 19 84 19 86 19 88 19 90 19 92 19 94 19 96 Business Inventories 19 98 20 00 US GDP 20 02 20 04

US Inventories as % of GDP

25.0% 20.0% 15.0% 10.0% 5.0% 0.0% 1984 1986 1988 1990 1992 1994

Year

1996 1998 2000 2002 2004 Source: CSCMP, Bureau of Economic Analysis

Two Questions

Two main Inventory Questions:  How much to buy?

 When is it time to buy?

Also: Which products to buy?

From whom?

Types of Inventory

 Raw Materials  Subcomponents  Work in progress (WIP)  Finished products  Defectives  Returns

Inventory Costs

What costs do we experience because we carry inventory?

Inventory Costs

Costs associated with inventory:  Cost of the products  Cost of ordering  Cost of hanging onto it  Cost of having too much / disposal  Cost of not having enough (shortage)

Shrinkage Costs

  How much is stolen?

   2% for discount, dept. stores, hardware, convenience, sporting goods 3% for toys & hobbies 1.5% for all else Where does the missing stuff go?

    Employees: 44.5% Shoplifters: 32.7% Administrative / paperwork error: 17.5% Vendor fraud: 5.1%

Inventory Holding Costs

Category Housing (building) cost Material handling Labor cost Opportunity/investment Pilferage/scrap/obsolescence Total Holding Cost % of Value 4% 3% 3% 9% 2% 21%

Inventory Models

  Fixed order quantity models  How much always same, when changes  Economic order quantity  Production order quantity  Quantity discount Fixed order period models  How much changes, when always same

Economic Order Quantity

Assumptions  Demand rate is known and constant  No order lead time  Shortages are not allowed  Costs:  S - setup cost per order  H - holding cost per unit time

EOQ

Inventory Level Q*

Optimal Order Quantity Decrease Due to Constant Demand

Time

EOQ

Inventory Level Q*

Optimal Order Quantity Instantaneous Receipt of Optimal Order Quantity

Time

EOQ

Q* Inventory Level Reorder Point (ROP) Lead Time Time

EOQ

Q* Inventory Level Reorder Point (ROP) Lead Time

Average Inventory Q/2

Time

Total Costs

 Average Inventory = Q/2  Annual Holding costs = H * Q/2  # Orders per year = D / Q  Annual Ordering Costs = S * D/Q  Cost of Goods = D * C  Annual Total Costs = Holding + Ordering + CoG

(

) 

2 

D S



How Much to Order?

Annual Cost Holding Cost = H * Q/2 Order Quantity

How Much to Order?

Annual Cost Ordering Cost = S * D/Q Holding Cost = H * Q/2 Order Quantity

How Much to Order?

Annual Cost Total Cost = Holding + Ordering Order Quantity

How Much to Order?

Annual Cost Total Cost = Holding + Ordering Optimal Q Order Quantity

Optimal Quantity

Total Costs = Take derivative with respect to Q = Solve for Q:

2 

DS Q

2 

S D



D H

2 

S D

2  0

Set equal to zero

 2

DS H Q

 2

DS H

Adding Lead Time

  Use same order size

Order before inventory depleted   2

DS H

 

= average demand rate (per day) L = lead time (in days)  both in same time period (wks, months, etc.)

A Question:

 If the EOQ is based on so many horrible assumptions that are never really true, why is it the most commonly used ordering policy?

 Profit function is very shallow  Even if conditions don’t hold perfectly, profits are close to optimal  Estimated parameters will not throw you off very far

Quantity Discounts

 How does this all change if price changes depending on order size?

 Holding cost as function of cost:  H = I * C  Explicitly consider price:

 2

DS I





Discount Example

D = 10,000 S = $20 I = 20% PriceQuantity EOQ c = 5.00

Q < 500 4.50

501-999 3.90

Q >= 1000 633 666 716

Discount Pricing

Total Cost

Price 1 Price 2 Price 3 500

633

666

716 1,000

Order Size

Discount Pricing

Total Cost

Price 1 Price 2 Price 3 500

633

666

716 1,000

Order Size

Discount Example

Order 666 at a time: Hold 666/2 * 4.50 * 0.2= $299.70

Order 10,000/666 * 20 = $300.00

Mat’l 10,000*4.50 = $45,000.00 45,599.70

Order 1,000 at a time: Hold 1,000/2 * 3.90 * 0.2=$390.00

Order 10,000/1,000 * 20 = $200.00

Mat’l 10,000*3.90 = $39,000.00

39,590.00

Discount Model

1. Compute EOQ for next cheapest price 2. Is EOQ feasible? (is EOQ in range?) If EOQ is too small, use lowest possible Q to get price.

3. Compute total cost for this quantity 4.

Repeat until EOQ is feasible or too big.

Select quantity/price with lowest total cost.

INVENTORY MANAGEMENT -- RANDOM DEMAND

Random Demand

 Don’t know how many we will sell  Sales will differ by period  Average always remains the same  Standard deviation remains constant

Impact of Random Demand

How would our policies change?

 How would our order quantity change?

 How would our reorder point change?

Mac’s Decision

      How many papers to buy?

Average = 90, st dev = 10 Cost = 0.20, Sales Price = 0.50

Salvage = 0.00

Cost of overestimating Demand, C O  C O = 0.20 - 0.00 = 0.20

Cost of Underestimating Demand, C U  C U = 0.50 - 0.20 = 0.30

Optimal Policy

G(x) = Probability demand <= x Optimal quantity: Pr(D  Q) 

C o C



u C u

Mac: G(x) = 0.3 / (0.2 + 0.3) = 0.6

From standard normal table, z = 0.253

=Normsinv(0.6) = 0.253

Q* = avg + z s = 90+ 2.53*10 = 90 +2.53 = 93

Optimal Policy

 If units are discrete, when in doubt, round up  If u units are on hand, order Q - u units  Model is called “newsboy problem,” newspaper purchasing decision  By time realize sales are good, no time to order more  By time realize sales are bad, too late, you’re stuck  Similar to the problem of # of Earth Day shirts to make, lbs. of Valentine’s candy to buy, green beer, Christmas trees, toys for Christmas, etc., etc.

Random Demand – Fixed Order Quantity  If we want to satisfy all of the demand 95% of the time, how many standard deviations above the mean should the inventory level be?

Probabilistic Models

Safety stock = x From statistics,

m



s  m

Safety stock Therefore, z =

L From normal table z .95

= 1.65

& Safety stock = z

L Safety stock = z

L R =

= 1.65*10 = 16.5

+ Safety Stock =350+16.5 = 366.5 ≈ 367

Random Example

    What should our reorder point be?

 demand over the lead time is 50 units,   with standard deviation of 20 want to satisfy all demand 90% of the time  (i.e., 90% chance we do not run out) To satisfy 90% of the demand, z = 1.28

Safety stock = z σ L = 1.28 * 20 = 25.6

R = 50 + 25.6 = 75.6

St Dev Over Lead Time

  What if we only know the average daily demand, and the standard deviation of daily demand?

 Lead time = 4 days,  daily demand = 10,  standard deviation = 5, What should our reorder point be, if z = 3?

St Dev Over LT

  If the average each day is 10, and the lead time is 4 days, then the average demand over the lead time must be 40.

 10 * 4  40 What is the standard deviation of demand over the lead time?

 Std. Dev. ≠ 5 * 4

St Dev Over Lead Time

 Standard deviation of demand = s

 

L days

day

4  5  10





 R = 40 + 3 * 10 = 70 



z L days

day

Service Level Criteria

  Type I: specify probability that you do not run out during the lead time  Probability that 100% of customers go home happy Type II: proportion of demands met from stock  Percentage that go home happy, on average  Fill Rate: easier to observe, is commonly used  G(z)= expected value of shortage, given z. Not frequently listed in tables

(

)  s

Q L

 1 

Fill Rate



Two Types of Service

Cycle Demand 1 180 2 3 4 75 235 140 10 Sum 5 6 7 8 9 180 200 150 90 160 40 1,450 Stock-Outs 0 0 45 0 0 10 0 0 0 0 55

Type I:

8 of 10 periods 80% service

Type II:

1,395 / 1,450 = 96%