INVENTORY MODELING

1
INVENTORY MODELING

• Items in inventory in a store
• Items waiting to be shipped
• Employees in a firm
• Computer information in computer files
• Etc.

2
COMPONENTS OF AN INVENTORY POLICY

• Q the amount to order (the order quantity)
• R when to reorder (the reorder point)

3
BASIC CONCEPT

• Balance the cost of having goods in inventory to
  other costs such as
• Order Cost
• Purchase Costs
• Shortage Costs

4
HOLDING COSTS

• Costs of keeping goods in inventory
• Cost of capital
• Rent
• Utilities
• Insurance
• Labor
• Taxes
• Shrinkage, Spoilage, Obsolescence

5
Holding Cost RateAnnual Holding Cost Per Unit

• These factors, individually are hard to determine
• Management (typically the CFO) assigns a holding
  cost rate, H, which is a percentage of the value
  of the item, C
• Annual Holding Cost Per Unit, Ch
• Ch HC (in /item in inv./year)

6
PROCUREMENT COSTS

• When purchasing items, this cost is known as the
  order cost, CO (in /order)
• These are costs associated with the ordering
  process that are independent of the size of the
  order-- invoice writing or checking, phone calls,
  etc.
• Labor
• Communication
• Some transportation

7
PROCUREMENT COSTS

• When these costs are associated with producing
  items for sale they are called set-up costs
  (still labeled CO-- in /setup)
• Costs associated with getting the process ready
  for production (regardless of the production
  quantity)
• Readying machines
• Calling in, shifting workers
• Paperwork, communications involved

8
PURCHASE/PRODUCTION COSTS

• These are the per unit purchase costs, C, if we
  are ordering the items from a supplier
• These are the per unit production costs, C, if we
  are producing the items for sale

9
CUSTOMER SATISFACTION COSTS

• Shortage/Goodwill Costs associated with being out
  of stock
• goodwill
• loss of future sales
• labor/communication
• Fixed administrative costs ? (/occurrence)
• Annualized Customer Waiting Costs
• Cs (/item short/year)

10
BASIC INVENTORY EQUATION

• (Total Annual Inventory Costs)
• (Total Annual Order/Setup-Up Costs)
• (Total Annual Holding Costs)
• (Total Annual Purchase/Production Costs)
• (Total Annual Shortage/Goodwill Costs)
• This is a quantity we wish to minimize!!

11
REVIEW SYSTEMS

• Continuous Review --
• Items are monitored continuously
• When inventory reaches some critical level, R, an
  order is placed for additional items
• Periodic Review --
• Ordering is done periodically (every day, week, 2
  weeks, etc.)
• Inventory is checked just prior to ordering to
  determine an order quantity

12
TIME HORIZONS

• Infinite Time Horizon
• Assumes the process has and will continue
  forever
• Single Period Models
• Ordering for a one-time occurence

13
EOQ-TYPE MODELS

• EOQ (Economic Order Quantity-type models assume
• Infinite Time Horizon
• Continuous Review
• Demand is relatively constant

14
THE BASIC EOQ MODEL

• Order the same amount, Q, each time
• Reordering is instantaneous
• Demand is relatively constant at D items/yr.
• Infinite Time Horizon/Continuous Review
• No shortages
• Since reordering is instantaneous

15
Economic Order Quantity
Figure 13.2
16
THE EOQ COST COMPONENTS

• Total Annual Order Costs
• (Cost/order)(average orders per year) CO(D/Q)
• Total Annual Holding Costs
• (Cost Per Item in inv./yr.)(Average inv.)
  Ch(Q/2)
• Total Annual Purchase Costs
• (Cost Per Item)(Average items ordered/yr.) CD

17
Economic Order Quantity
Figure 13.3
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THE EOQ TOTAL COST EQUATION

• TC(Q) CO(D/Q) Ch(Q/2) CD
• This a function in one unknown (Q) that we wish
  to minimize

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SOLVING FOR Q

• TC(Q) CO(D/Q) Ch(Q/2) CD

20
THE REORDER POINT, r

• Since reordering is instantaneous, r 0
• MODIFICATION -- fixed lead time L yrs.
• r LD
• But demand was only approximately constant so we
  may wish to carry some safety stock (SS) to
  lessen the likelihood of running out of stock
• Then, r LD SS

21
TOTAL ANNUAL COST

• The optimal policy is to order Q when supply
  reaches r
• TC(Q) COD/Q (Ch/2)(Q) CD ChSS
• ltvariable costgt fixed safety
• cost stock cost
• The optimal policy minimizes the total variable
  cost, hence the total annual cost

22
TOTAL VARIABLE COST CURVE

• Ignoring fixed costs and safety stock costs

23
EXAMPLE -- ALLEN APPLIANCE COMPANY

• Juicer Sales For Past 10 weeks
• 1. 105 6. 120
• 2. 115 7. 135
• 3. 125 8. 115
• 4. 120 9. 110
• 5. 125 10. 130
• Using 10-period moving average method,
• D (105 115 130)/10 120/ wk
  6240/yr

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ALLEN APPLIANCE COSTS

• Juicers cost 10 each and sell for 11.85
• Cost of money 10
• Other misc. costs associated with inventory 4
• Labor, postage, telephone charges/order 8
• Workers paid 12/hr. -- 20 min. to unload an
  order
• H .10 .04 .14 Ch .14(10) 1.40
• CO 8 (1/3 hr.)(12/hr.) 8 4 12

25
OPTIMAL ORDER QUANTITY FOR ALLEN

26
OPTIMAL QUANTITIES

• Total Order Cost COD/Q (12)(6240)/327
  228.99
• Total Holding Cost (Ch/2)Q
  (1.40/2)(327)
  228.90
• (Total Order Cost Total Holding Cost -- except
  for roundoff)
• Orders Per Year D/Q 6240/327 19.08
• Time between orders (Cycle Time)
  Q/D 327/6240 .0524 years 2.72
  weeks

27
TOTAL ANNUAL COST

• Total Variable Cost Total Order Cost Total
  Holding Cost 228.99 228.90 457.89
• Total Fixed Cost CD 10(6240) 62,400
• Total Annual Cost 457.89 62,400
  62,857.89

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WHY IS EOQ MODEL IMPORTANT?

• No real-life model really is an EOQ model
• Many models are variants of EOQ-type models
• Many situations can be approximated by EOQ models
• The EOQ model is relatively insensitive to some
  pretty major errors in input parameters

29
INSENSIVITY IN EOQ MODELS

• We cannot affect fixed costs, only variable costs
• TV(Q) COD/Q (Ch/2)(Q)
• Now, suppose D really 7500 (gt20 error)
• We did not know this and got Q 327
• TV(327) ((12)(7500))/327 (1.40/2)(327)
  504.13
• Q should have been SQRT(2(12)(7500)/1.40)
  359
• TV(359) ((12)(7500))/359 (1.40/2)(359)
  502.00
• This is only a 0.4 increase in the TVCost

30
DETERMINING A REORDER POINT, r (Without Safety
Stock)

• Suppose lead time is 8 working days
• The company operates 260 days per year
• r LD where L and D are in the same time units
• L 8/260 ? .0308 yrs D 6240 /year
• r .0308(6240) ? 192
• OR,
• L 8 days D/day 6240/260 24
• r 8(24) 192

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ACTUAL DEMAND DISTRIBUTION

• Suppose we can assume that demand follows a
  normal distribution
• This can be checked by a goodness of fit test
• From our data, over the course of a week, W, we
  can approximate ?W by (105 130) 120
• ?W2 ? sW2 ((1052 1302) - 10(120)2)/9 ? 83.33

32
DEMAND DISTRIBUTION DURING 8 -DAY LEAD TIME

• Normal
• 8 days 8/5 1.6 weeks, so
• ?L (1.6)(120) 192
• ?L2 ? (1.6)(83.33) 133.33
• ?L ?

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SAFETY STOCK

• Suppose we wish a cycle service level of 99
• WE wish NOT to run out of stock in 99 of our
  inventory cycles
• Reorder point, r ?L z.01 ?L
• 192 2.33(11.55) ? 219
• 219 - 192 27 units safety stock 2.33(11.55)
  
• Safety stock cost ChSS 1.40(27) 37.80
• This should be added to the TOTAL ANNUAL COST

34
OTHER EOQ-TYPE MODELS

• Quantity Discount Models
• Production Lot Size Models
• Planned Shortage Model
• ALL SEEK TO MINIMIZE THE TOTAL ANNUAL COST
  EQUATION

35
QUANTITY DISCOUNTS

• All-units vs. incremental discounts
• ALL UNITS DISCOUNTS FOR ALLEN
• Quantity Unit Cost
• lt 300 10.00
• 300-600 9.75
• 600-1000 9.50
• 1000-5000 9.40
• ?5000 9.00

36
PIECEWISE APPROACH

• For each piece of the total cost equation, the
  minimum cost for the piece is at an end point or
  at its Q
• If Q for a piece lies
• above the upper interval limit -- ignore this
  piece
• within this piece -- it is optimal for this piece
• below the lower interval limit -- the lower
  interval limit is optimal for this piece
• Calculate the total annual cost using the best
  value for Q for each piece, and choose the lowest

37
QUANTITY DISCOUNT APPROACH FOR ALLEN

• When C changes, only Ch changes in the formula
  for Q since Ch .14C
• Quantity Unit Cost Ch Q Best Q
  TC
• lt 300 10.00 1.40 327 ----
  ----
• 300-600 9.75 1.365 331 331
  61,292
• 600-1000 9.50 1.33 336 600
  59,804
• 1000-5000 9.40 1.316 337 1000
  59,389
• ?5000 9.00 1.26 345 5000 59,325
• ORDER 5000

38
OTHER CONSIDERATIONS

• 5000 is 5000/6240 .8 years 9.6 months supply
• May not wish to order that amount
• Company policy may be DO NOT ORDER MORE THAN A
  3-MONTHS SUPPLY 6240/4 1560
• If that is the case, since 1560 is in the
  interval from 1000 - 5000 and the best Q in that
  interval is 1000, 1000 should be ordered

39
PRODUCTION LOT SIZE PROBLEMS

• We are producing at a rate P/yr. That is greater
  than the demand rate of D/yr.
• Inventory does not jump to Q but builds up to a
  value IMAX that is reached when production is
  ceased
• Length of a production time Q/P
• IMAX P(Q/P) - D(Q/P) (1-D/P)Q
• Average inventory IMAX/2 ((1-D/P)/2)Q

40
Economic Production Quantity
On-hand Inventory
Time
Figure G.1
41
Economic Production Quantity
Production quantity
Q
On-hand Inventory
Time
Figure G.1
42
Economic Production Quantity
Production quantity
Q
Demand during production interval
On-hand Inventory
p - d
Time
Figure G.1
43
Economic Production Quantity
Production quantity
Q
Demand during production interval
On-hand Inventory
p - d
Time
Figure G.1
44
Economic Production Quantity
Production quantity
Q
Demand during production interval
On-hand Inventory
p - d
Time
Production and demand
Demand only
TBO
Figure G.1
45
Economic Production Quantity
Production quantity
Q
Demand during production interval
On-hand Inventory
p - d
Time
Production and demand
Demand only
TBO
Figure G.1
46
Economic Production Quantity
Production quantity
Q
Demand during production interval
Imax
On-hand Inventory
Maximum inventory
p - d
Time
Production and demand
Demand only
TBO
Figure G.1
47
PRODUCTION LOT SIZE -- TOTAL ANNUAL COST

• CO Set-up cost rather than order cost
• Set-up time for production ? lead time
• Q The production lot size
• TC(Q) CO(D/Q) Ch((1-D/P)/2)Q CD

48
OPTIMAL PRODUCTION LOT SIZE, Q

• TC(Q) CO(D/Q) Ch((1-D/P)/2)Q CD

49
EXAMPLE-- Farah Cosmetics

• Production Capacity 1000 tubes/hr.
• Daily Demand 1680 tubes
• Production cost 0.50/tube (C 0.50)
• Set-up cost 150 per set-up (CO 150)
• Holding Cost rate 40 (Ch .4(.50) .20)
• D 1680(365) 613,200
• P 1000(24)(365) 8,760,000

50
OPTIMAL PRODUCTION LOT SIZE

51
TOTAL ANNUAL COST

• TOTAL ANNUAL COST
• TV(Q) CO(D/Q) Ch((1-D/P)/2)Q
• (150)(613,200/31,449)
• .2(1-613,200/8,760,00)(31,449) 5,850
• TC(Q) TV(Q) CD
• 5,850 .50(613,200) 312,450

52
OTHER QUANTITES

• Length of a Production run Q/P
• 31,449/8,760,000 .00359yrs. .00359(365)
• 1.31 days
• Length of a Production cycle Q/D
• 31,449/613,200 .0512866yrs. .00512866(365)
• 18.72 days
• Number of Production runs/yr. D/Q 19.5
• IMAX (1-613,200/8,760,00)(31,449) 29,248

53
PLANNED SHORTAGE MODEL

• Assumes no customers will be lost because of
  stockouts
• Stockout costs
• ? -- fixed administrative cost/stockout
• Cs -- annualized cost per unit short
• Acts like a holding cost in reverse
• We plan on being short by S items when an order
  of size Q comes in

54
PROPORTION OF TIME OUT OF STOCK

• T1 time of a cycle with inventory
• T2 time of a cycle out of stock
• T T1 T2 time of a cycle
• IMAX Q-S
• Proportion of time in stock T1/T (Q-S)/Q
• Proportion of time out of stock T2/T S/Q
• Avg. inventory ((Q-S)/Q)((Q-S)/2) (Q-S)2/2Q
• Average Stockouts (S/Q)(S/2) S2/2Q

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TOTAL ANNUAL COST EQUATION

• TC(Q,S) CO(D/Q) Ch((Q-S)2/2Q) ?S(D/Q)
  Cs(S2/2Q) CD
• Take partial derivatives with respect to Q and S
  and set 0. We get two equations in the two
  unknowns Q and S.

56
OPTIMAL ORDER QUANTITY, QOPTIMAL BACKORDERS,
S

57
EXAMPLESCANLON PLUMBING

• Saunas cost 2400 each (C 2400)
• Order cost 1250 (CO 1250)
• Holding Cost 525/unit /yr. (Ch 525)
• Backorder Good will Cost 20/wk (CS 1040)
• Backorder Admin. Cost 10/order (? 10)
• Demand 15/wk (D780)

58
RESULTS

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What IF Lead Time Were 4 Weeks?

• Demand over 4 weeks 4(15) 60
• Want order to arrive when there are 20
  backorders.
• Thus order should be placed when there are 60 -
  20 40 saunas left in inventory

60
Part III Single-Period Model Newsvendor

• Used to order perishables or other items with
  limited useful lives.
• Fruits and vegetables, Seafood, Cut flowers.
• Blood (certain blood products in a blood bank)
• Newspapers, magazines,
• Unsold or unused goods are not typically carried
  over from one period to the next rather they are
  salvaged or disposed of.
• Model can be used to allocate time-perishable
  service capacity.
• Two costs shortage (short) and excess (long).

61
Single-Period Model

• Shortage or stockout cost may be a charge for
  loss of customer goodwill, or the opportunity
  cost of lost sales (or customer!)
• Cs Revenue per unit - Cost per unit.
• Excess (Long) cost applies to the items left over
  at end of the period, which need salvaging
• Ce Original cost per unit - Salvage value per
  unit.
• (insert smoke, mirrors, and the magic of
  Leibnitzs Rule here)

62
The Single-Period Model Newsvendor

• How do I know what service level is the best one,
  based upon my costs?
• Answer Assuming my goal is to maximize profit
  (at least for the purposes of this analysis!) I
  should satisfy SL fraction of demand during the
  next period (DDLT)
• If Cs is shortage cost/unit, and Ce is excess
  cost/unit, then

63
Single-Period Model for Normally Distributed
Demand

• Computing the optimal stocking level differs
  slightly depending on whether demand is
  continuous (e.g. normal) or discrete. We begin
  with continuous case.
• Suppose demand for apple cider at a downtown
  street stand varies continuously according to a
  normal distribution with a mean of 200 liters per
  week and a standard deviation of 100 liters per
  week
• Revenue per unit 1 per liter
• Cost per unit 0.40 per liter
• Salvage value 0.20 per liter.

64
Single-Period Model for Normally Distributed
Demand

• Cs 60 cents per liter
• Ce 20 cents per liter.
• SL Cs/(Cs Ce) 60/(60 20) 0.75
• To maximize profit, we should stock enough
  product to satisfy 75 of the demand (on
  average!), while we intentionally plan NOT to
  serve 25 of the demand.
• The folks in marketing could get worried! If
  this is a business where stockouts lose long-term
  customers, then we must increase Cs to reflect
  the actual cost of lost customer due to stockout.

65
Single-Period Model for Continuous Demand

• demand is Normal(200 liters per week, variance
  10,000 liters2/wk) so ? 100 liters per week
• Continuous example continued
• 75 of the area under the normal curve must be to
  the left of the stocking level.
• Appendix shows a z of 0.67 corresponds to a
  left area of 0.749
• Optimal stocking level mean z (?) 200
  (0.67)(100) 267. liters.

66
Single-Period Discrete Demand Lively Lobsters

• Lively Lobsters (L.L.) receives a supply of
  fresh, live lobsters from Maine every day. Lively
  earns a profit of 7.50 for every lobster sold,
  but a day-old lobster is worth only 8.50. Each
  lobster costs L.L. 14.50.
• (a) what is the unit cost of a L.L. stockout?
• Cs 7.50 lost profit
• (b) unit cost of having a left-over lobster?
• Ce 14.50 - 8.50 cost salvage value 6.
• (c) What should the L.L. service level be?
• SL Cs/(Cs Ce) 7.5 / (7.5 6) .56
  (larger Cs leads to SL gt .50)
• Demand follows a discrete (relative frequency)
  distribution as given on next page.

67
Lively Lobsters SL Cs/(Cs Ce) .56

• Demand follows a discrete (relative frequency)
  distribution
• Result order 25 Lobsters, because that is the
  smallest amount that will serve at least 56 of
  the demand on a given night.