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INVENTORY MODELING
• Items in inventory in a store
• Items waiting to be shipped
• Employees in a firm
• Computer information in computer files
• Etc.
COMPONENTS OF AN INVENTORY POLICY
• Q = the amount to order (the order quantity)
• R = when to reorder (the reorder point)
BASIC CONCEPT
• Balance the cost of having goods in inventory to other costs such as:– Order Cost– Purchase Costs– Shortage Costs
HOLDING COSTS
• Costs of keeping goods in inventory– Cost of capital– Rent– Utilities– Insurance– Labor– Taxes– Shrinkage, Spoilage, Obsolescence
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)
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
PROCUREMENT COSTS
• When these costs are associated with producing items for sale they are called set-up costs (still labeled CO-- in $/setup)
• Costs associated with getting the process ready for production (regardless of the production quantity)– Readying machines
– Calling in, shifting workers
– Paperwork, communications involved
PURCHASE/PRODUCTION COSTS
• These are the per unit purchase costs, C, if we are ordering the items from a supplier
• These are the per unit production costs, C, if we are producing the items for sale
CUSTOMER SATISFACTION COSTS
• Shortage/Goodwill Costs associated with being out of stock– goodwill– loss of future sales– labor/communication
• Fixed administrative costs = ($/occurrence)• Annualized Customer Waiting Costs =
Cs ($/item short/year)
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!!
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
TIME HORIZONS
• Infinite Time Horizon– Assumes the process has and will continue
“forever”
• Single Period Models – Ordering for a one-time occurence
EOQ-TYPE MODELS
• EOQ (Economic Order Quantity-type models assume:
• Infinite Time Horizon
• Continuous Review
• Demand is relatively constant
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
Economic Order QuantityEconomic Order QuantityO
n-h
and
in
ven
tory
(u
nit
s)
Time
Averagecycleinventory
Q
Q—2
1 cycle
Receive order
Inventory depletion (demand rate)
Figure 13.2
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
Economic Order QuantityEconomic Order QuantityA
nn
ual
co
st
(do
llars
)
Lot Size (Q)
Ordering cost
Holding cost
Total cost = Holding Cost + Set-up Cost
Figure 13.3
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
SOLVING FOR Q*
• TC(Q) = CO(D/Q) + Ch(Q/2) + CD
Formula EOQ The 2
2
,
002
*
2
2
h
O
h
O
hO
C
DCQ
C
DCQ
Solving
C
Q
DC
dQ
dTC
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
TOTAL ANNUAL COST
• The optimal policy is to order Q* when supply reaches r*
• TC(Q*) = COD/Q* + (Ch/2)(Q*) + CD + ChSS
<==variable cost==> fixed safety
cost stock cost
• The optimal policy minimizes the total variable cost, hence the total annual cost
TOTAL VARIABLE COST CURVE
• Ignoring fixed costs and safety stock costs:
• The Total Variable Costs function
Constructing the Total Annual Variable Cost CurveAdd the two curves to one another
* * o * * *Total Annual Holding and Ordering Costs
Q
TV(Q)
Q*
The optimal order size
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
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
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
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
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
INSENSIVITY IN EOQ MODELS
• We cannot affect fixed costs, only variable costsTV(Q) = COD/Q + (Ch/2)(Q)
Now, suppose D really = 7500 (>20% 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
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
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 sW
2 = ((1052 +…+1302) - 10(120)2)/9 83.33
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 55.1133.133
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
OTHER EOQ-TYPE MODELS
• Quantity Discount Models
• Production Lot Size Models
• Planned Shortage Model
ALL SEEK TO MINIMIZE THE TOTAL ANNUAL COST EQUATION
QUANTITY DISCOUNTS
• All-units vs. incremental discounts
ALL UNITS DISCOUNTS FOR ALLEN
Quantity Unit Cost
< 300 $10.00
300-600 $ 9.75
600-1000 $ 9.50
1000-5000 $ 9.40
5000 $ 9.00
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
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
< 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
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
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
Economic Production QuaEconomic Production Quantityntity
Production quantity
On
-han
d I
nve
nto
ry
Q
Time
Figure G.1
Economic Production QuaEconomic Production Quantityntity
Production quantity
On
-han
d I
nve
nto
ry
Q
Time
Figure G.1
Demand during production interval
p - d
Economic Production QuaEconomic Production Quantityntity
Production quantity
Demand during production interval
On
-han
d I
nve
nto
ry
Q
Time
p - d
Figure G.1
Economic Production QuaEconomic Production Quantityntity
Production quantity
Demand during production interval
On
-han
d I
nve
nto
ry
Q
Time
p - d
Figure G.1
Production and demand
Demand only
TBO
Economic Production QuaEconomic Production Quantityntity
Production quantity
Demand during production interval
Production and demand
Demand only
TBO
On
-han
d I
nve
nto
ry
Q
Time
p - d
Figure G.1
Economic Production QuaEconomic Production Quantityntity
Production quantity
Demand during production interval
Maximum inventory
Production and demand
Demand only
TBO
On
-han
d I
nve
nto
ry
Q
Time
IImaxmax
p - d
Figure G.1
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
OPTIMAL PRODUCTION LOT SIZE, Q*
• TC(Q) = CO(D/Q) + Ch((1-D/P)/2)Q + CD
SizeLot )/1(
2
)/1(
2
,
002
)/1(
*
2
2
PDC
DCQ
PDC
DCQ
Solving
PDC
Q
DC
dQ
dTC
h
O
h
O
hO
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
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
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
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
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
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.
*Q
OPTIMAL ORDER QUANTITY, Q*OPTIMAL # BACKORDERS, S*
sh
h
shs
sh
h
O
CC
DQCS
CC
D
C
CC
C
DCQ
**
)(2*
2
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 (D=780)
RESULTS
backorders20 are e when ther74order Re
201040525
)10)(780()74)(525(*
74)1040)(525(
)10*780(
1040
1040525
525
)780)(1250(2*
2
S
Q
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
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).
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 Leibnitz’s Rule here…)
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
SLC
C Cs
s e
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.
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.
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.
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 > .50)
• Demand follows a discrete (relative frequency) distribution as given on next page.
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.
Probability that demand
Demand
Relative Frequency
(pmf)
Cumulative Relative
Frequency (cdf)
will be less than or equal to x
19 0.05 0.05 P(D < 19 )
20 0.05 0.10 P(D < 20 )
21 0.08 0.18 P(D < 21 )
22 0.08 0.26 P(D < 22 )
23 0.13 0.39 P(D < 23 )
24 0.14 0.53 P(D < 24 )
25 0.10 0.63 P(D < 25 )
26 0.12 0.75 P(D < 26 )
27 0.10 you do P(D < 27 )
28 0.10 you do P(D < 28 )
29 0.05 1.00 P(D < 29 )
* pmf = prob. mass function