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MULTI-PERIOD MODELS: PERFORMANCE MEASURES

LM6001 Inventory

management

1

TODAY’S AGENDA

Given the system parametersContinuous review (s,Q): Lot size, reorder point (ROP)

Periodic review (R,S): Order-up-to level (OUTL)

Determine key performance measures, e.g.,

Average inventory

Average backorder

Expected service level

Fill rate

In-stock probability

Annual average cost

2

3

Continuous (s,Q) Periodic (R,S)

Effective demand, 𝐷𝑒 𝐷𝐿 𝐷𝐿+𝑅

(𝜇𝑒, 𝜎𝑒) (𝜇1𝐿, 𝜎1 𝐿 ) ( 𝜇1(𝐿 + 𝑅), 𝜎1 (𝐿 + 𝑅) )

Performance measure 𝑧 = (𝑠 − 𝜇𝑒)/𝜎𝑒 𝑧 = (𝑆 − 𝜇𝑒)/𝜎𝑒

(Expected) order size 𝑄 ത𝑄 = 𝜇1𝑅 = 𝐸 𝐷𝑅

Cycle stock (Order size)/2

SS = E[IL at end of period (just before

replenishment arrives)]

𝐸 𝑠 − 𝐷𝑒 = 𝑠 − 𝜇𝑒 𝐸 𝑆 − 𝐷𝑒 = 𝑆 − 𝜇𝑒

E[Backorder at end of period], ത𝐵 𝐸 𝐷𝑒 − 𝑠 + = 𝜎𝑒𝐿(𝑧) 𝐸 𝐷𝑒 − 𝑆 + = 𝜎𝑒𝐿(𝑧)

E[On-hand inv at end of period] 𝐸 𝑠 − 𝐷𝑒+ = 𝑆𝑆 + ത𝐵 ≈ 𝑆𝑆 𝐸 𝑆 − 𝐷𝑒

+ = 𝑆𝑆 + ത𝐵 ≈ 𝑆𝑆

E[On-hand avg inv] = (Beginning+Ending)/2 (Cycle stock) + E[On-hand inv at end of period]

Cycle SL, in-stock probability 𝑃(𝐷𝑒 ≤ 𝑠) 𝑃(𝐷𝑒 ≤ 𝑆)

Fill rate,

1 −E[Backorder in one cycle]

E[Demand during one cycle]

1 −ത𝐵

𝑄1 −

ത𝐵

ത𝑄= 1 −

ത𝐵

𝜇1𝑅

(Expected) order frequency 𝐸 𝐷1 /𝑄 1 − 𝑃(𝐷𝑅 ≤ 0)

𝑅

Decision rule 𝑠 = 𝜇𝑒 + 𝑘 𝜎𝑒 𝑆 = 𝜇𝑒 + 𝑘 𝜎𝑒

Cycle SL, a 𝑘 = Φ−1(𝛼)

Fill rate𝑘 = 𝐿−1

1 − 𝛽 𝑄

𝜎𝑒𝑘 = 𝐿−1

1 − 𝛽 (𝜇1𝑅)

𝜎𝑒

4

• A cycle is defined as the duration which two successive replenishment orders are received.

• Effective demand is the demand during lead time (DDLT)

E[on-hand at end of cycle]

SAFETY STOCK

• Recall Inventory on-order, IO: # of units that we ordered in previous periods

that we have not yet received.

𝐼𝑃 = 𝐼𝐿 + 𝐼𝑂

Inventory on-hand

𝐼 = 𝐼𝐿 + = max 𝐼𝐿, 0

Backorder: Total amount of demand that has occurred but has not been satisfied

𝐵 = 𝐼𝐿 − = max(−𝐼𝐿, 0)

Safety stock is defined as the average inventory level at the end of cycle

IL = 𝐼𝐿 + - 𝐼𝐿 −

(Avg IL at the end) = (Avg On-hand at the end) – (Avg Backorder at end)

SS = (Avg on-hand) – (Avg backorder)

Thus, Avg on-hand = SS + (Avg Back order) 5

• Recall Inventory on-hand

𝐼 = 𝐼𝐿 + = max 𝐼𝐿, 0

Backorder: Total amount of demand that has occurred but has not been satisfied

𝐵 = 𝐼𝐿 − = max(−𝐼𝐿, 0)

Safety stock is defined as the average inventory level at the end of cycle

IL = 𝐼𝐿 + - 𝐼𝐿 −

(Avg IL at the end) = (Avg On-hand at the end) – (Avg Backorder at end)

SS = (Avg on-hand) – (Avg backorder)

Thus, Avg on-hand = SS + (Avg Back order)

6

Cycle 1 2 3 4 5 6 7 8 9 10 AvgIL at end of cycle 18 7 -4 12 10 -6 3 1 -9 8 4.0

on-hand 18 7 0 12 10 0 3 1 0 8 5.9backorder 0 0 4 0 0 6 0 0 9 0 1.9

AVERAGE OF A FUNCTION AVERAGE ON-HAND INV

7

• Cycle service level, or in-stock probability, is the probability that we are in stock in each cycle.

𝑃(𝐷𝐿 ≤ 𝑅𝑂𝑃)• Fill rate is the expected fraction of demand served immediately from

stock.

1 −ത𝐵

𝐸 𝑑𝑒𝑚𝑎𝑛𝑑 𝑖𝑛 𝑜𝑛𝑒 𝑐𝑦𝑙𝑒= 1 −

ത𝐵

𝑄

Over 10 cycles:• 2 out of 10 result in stockout; so,

the in-stock probability is 8/10=80%

• Total demand is 907, and the backorder is 27, so the demand served from stock is (907-27), and the fill rate is 907 − 27

907= 1 −

27

907= 97.03%

8

ROP

9

10

ROP Lot size (Q)

Cycle stock

Safety stock

Average backorder at end of cycle

Average on-hand inventory over time

Cycle service level (in-stock probability)

Fill rate

Order frequency

Average backorder cost

Average holding cost

Average setup cost

• Originally stock recording systems used to include stock control levels as “minimum” and “maximum” stock levels.

• The use of a minimum stock level for order control is not sensible, since the minimum occurs immediately before delivery.

• Items have to be ordered will in advance of this, so control is through a reorder point, not a minimum stock.

• However, a minimum stock level is vital to ensure that there is warning of low stocks.

• In a stock control system, there is a need for both reorder point and minimum stock level, i.e., safety stock.

11

R = Review period R S = Order-up-to level (OUTL) Effective demand is the demand during lead time plus review

R R

S

12

• Continuous review• Cycle service level, or in-stock probability, is the probability that we

are in stock in each cycle. 𝑃(𝐷𝐿 ≤ 𝑅𝑂𝑃)

• Fill rate is the expected fraction of demand served immediately from stock.

1 −ത𝐵

𝐸 demand in one cyle= 1 −

ത𝐵

𝑄

Periodic review In-stock probability is the probability that we are in stock in each

cycle. 𝑃 𝐷𝐿+𝑅 ≤ 𝑂𝑈𝑇𝐿

Fill rate is the expected fraction of demand served immediately from stock.

1 −ത𝐵

𝐸 demand during review period= 1 −

ത𝐵

𝜇1𝑅

13

Weekly demand is normally distributed

Mean of weekly demand 𝜇1 = 85

Standard deviation of weekly demand

𝜎1 = 17.5 Lead time L=3 weeks

1. Find ROP for 90% CSL2. Find ROP for 96% FR. Suppose lot size Q=200

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Effective demand• Mean, AVGL, 𝜇𝐿 = 𝜇1𝐿 = 85 3 = 255

• Standard deviation, STDL, 𝜎𝐿 = 𝜎1 𝐿 = 17.5 3 = 30.31

Given 90% CSL, 𝛼 = 0.90• Safety factor k=Φ−1(0.90)=1.28• SS = 𝑘 𝜎𝐿=1.28(30.31)=38.84• ROP = 𝜇𝐿 + 𝑆𝑆=255+38.84=293.84 ≈294

Given 96% FR, 𝛽 = 0.96.

• Safety factor k=L−11−𝛽 𝑄

𝜎𝐿= L−1

1−0.96 200

30.31= L−1(0.2639)=0.31

• SS = 𝑘 𝜎𝐿=0.31(30.31)=9.40• ROP = 𝜇𝐿 + 𝑆𝑆=255+9.40=264.6 ≈265

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• ROP, s =278. Then, 𝑧 =𝑠−𝜇𝐿

𝜎𝐿=

278−255

30.31= 0.76. L(z) = L(0.76)= 0.129184

• Lot size, Q = 200. Then Cycle stock (avg inventory) = Q/2 = 200/2 = 100. • SS = 𝑠 − 𝜇𝐿 = 278 − 255 = 23• E[backorder at end of cycle] ത𝐵 = 𝜎𝐿𝐿 𝑧 = (30.31)(0.129184) = 3.92 units• E[on-hand inv] = (cycle stock) + SS + ത𝐵 = 100+ 23 + 3.92 =126.92 units• E[in-transit inv] = ((85)(52))(3/52) = 𝜇𝐿= 255 units (Recall Little’s Law 𝐿 = 𝜆𝑊 where L is avg # of

customers in system, 𝜆 avg no of arrivals entering the system, W avg time a customer spends in system)

• Order freq = 𝜇1/𝑄=85/200=0.4250 times per week (or 48*0.4250=20.4 times/yr)• CSL = Φ 𝑧 = Φ(0.76)=0.776=77.6%• FR = 1 − ത𝐵 /Q = 1-3.9758/200=0.9801 = 98.01%



Standard deviation of weekly demand 𝜎1 = 17.5 Lead time L=3 weeks. (1 year = 48 weeks) Suppose Q=200, s=ROP=278. Calculate

1. Avg on-hand inventory, in-transit inventory2. Avg backorder3. Order frequency4. Service level: CSL, FR

Effective demand• Mean, AVGL, 𝜇𝐿 = 𝜇1𝐿 = 85 3 = 255

• Standard deviation, STDL, 𝜎𝐿 = 𝜎1 𝐿 = 17.5 3 = 30.31

1. Given 90% CSL, find safety stock and ROP. 2. Suppose that the order quantity the company is using for this

SKU is 500 units. The company wants to maintain 90% CSL. a) What is the average inventory level? b) What is the average holding cost per year? Also average setup

cost per year 3. Suppose that the lead time is reduced to 3 weeks. Assume

order quantity of 50 and CSL of 90%. Repeat problem 1 and 2. 4. The order quantity the company is using for this SKU is 50

units Suppose lead time is 6 weeks. You want 99% SL. What is the average inventory level?

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Ordering cost (THB/order) 500.00

Holding cost factor (/Year) 27%

Unit cost (THB/unit) 324.00

Unit Holding cost (THB/unit/Yr) 87.48

Assume that 1 year = 12 months = 48 weeks Monthly demand is normally distributed with

mean 85 units and standard deviation 26 units. The replenishment lead time is 6 weeks.

1. Given 90% CSL, find safety stock and ROP. 2. Suppose that the order quantity the company is using for this SKU is

10,000 units. The company wants to maintain 90% CSL. [Ans: ROP=169]a) What is the average inventory level? [Ans:1.44 + 41.51 + 500/2 = 292.54]b) What is the average holding cost per year? Average setup cost per year = 500*((12*85)/500)=500*(2.04)=1020 THB/yr

3. Suppose that the lead time is reduced to 3 weeks. Assume order quantity of 50 and CSL of 90%. Repeat problem 1 and 2.

4. The order quantity the company is using for this SKU is 50 units Suppose lead time is 6 weeks. You want 99% SL. What is the average inventory level?

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Assume that 1 year = 12 months = 48 weeks Monthly demand is normally distributed with

mean 85 units and standard deviation 26 units. The replenishment lead time is 6 weeks.

If you wish to divide demand distribution from a long period length (e.g., a month) into n short periods (e.g., a week), then

E[demand in the short period] = E[demand in the long period]/n

Stdev of demand in short period = (Stdev of demand in long period)/ 𝑛

If you wish to combine a demand distribution from n short period lengths (e.g., a week ) into one long period (e.g., a three-week period), then

E[demand in the long period] = E[demand in the long period]*n

Stdev of demand in long period = (Stdev of demand in long period)* 𝑛

These assume demands in each period are independent and identically distributed.

Monthly demand is normally distributed with mean 85 units and standard deviation 26 units.

The replenishment lead time is 6 weeks.

Find mean and standard deviation of demand during lead time. (1 month = 4 weeks)

Sol 1 LT = 6 wks = 6/4 = 1.5 month 𝜇𝐿 = 85 1.5 = 127.5 units

𝜎𝐿 = 26 1.5 = 31.84 units Sol 2

Demand in one week has mean 85/4=21.25 units/week, standard deviation 26/ 4=13 units/week.

𝜇𝐿 = 21.25(6) = 127.5 units

𝜎𝐿 = 13 6 = 31.84 units

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19

• Desired CSL 𝛼 = 0.90• Safety factor k=Φ−1(0.90)=1.28• SS = 𝑘 𝜎𝐿+𝑅=1.28(39.13)=50.15• OUTL = 𝜇𝐿+𝑅 + 𝑆𝑆=425+ 50.15=475.15≈476

• Desired FR 𝛽 = 0.96.

• Safety factor k=L−11−𝛽 𝐸[ 𝑜𝑟𝑑𝑒𝑟 𝑠𝑖𝑧𝑒 ]

𝜎𝐿+𝑅= L−1(0.173774426)=0.58

• SS = 𝑘 𝜎𝐿+𝑅=0.58(39.13)=22.70• OUTL = 𝜇𝐿+𝑅 + 𝑆𝑆=425+22.70=447.70 ≈448



Standard deviation of weekly demand 𝜎1 = 17.5

Periodic review with lead time L=3 weeks and review period R=2 weeks

1. Given 90% CSL, find OUTL2. Given 96% FR, find OUTL

Effective demand• Mean, AVGLR, 𝜇𝐿+𝑅 = 𝜇1 𝐿 + 𝑅 = 85 5 =425

• Standard deviation, STDLR, 𝜎𝐿+𝑅 = 𝜎1 (𝐿 + 𝑅) = 17.5 5= 39.13

• Expected order size = 𝜇1𝑅=(85)(2)=170



Standard deviation of weekly demand 𝜎1 = 17.5

Periodic review with lead time L=3 weeks and review period R=2 weeks

Given OUTL, S=450, find performance measures

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Effective demand• Mean, AVGLR, 𝜇𝐿+𝑅 = 𝜇1 𝐿 + 𝑅 = 85 5 =425

• Standard deviation, STDLR, 𝜎𝐿+𝑅 = 𝜎1 (𝐿 + 𝑅) = 17.5 5= 39.13• Expected order size = 𝜇1𝑅=(85)(2)=170

• Expected order size = 𝜇1𝑅=(85)(2)=170

• Then, 𝑧 =𝑆−𝜇𝐿+𝑅

𝜎𝐿+𝑅=(450-425)/39.13=0.64 L(z) = L(0.64)=0.160594

• Cycle stock = E(order size)/2 = 170/2 = 85. • SS = S−𝜇𝐿+𝑅=450-425=25• E[backorder at end of cycle] ത𝐵 = 𝜎𝐿+𝑅𝐿 𝑧 =(39.13)(0.160594)=6.2842• E[on-hand avg inv] = (cycle stock) + SS + ത𝐵 = 85+ 25 + 6.2842 = =116.2842• CSL = Φ 𝑧 = Φ(0.64)=0.7385=73.85%• FR = 1- ത𝐵 /E(order size) =1- 6.2842/170=0.9630=96.30%

• Order frequency =(1/R)P(𝐷𝑅 > 0) = (1/(2/48))*[1 − Φ(−170/(17.5 2))]=24*[1-Φ(−6.869)] = 24 times/year

1. Given 90% CSL, find safety stock and OUTL. 2. The company wants to maintain 90% CSL.

a) What is the average inventory level? b) What is the observed fill rate? c) What is the average annual cost?

3. Suppose that the length of the review period time increases to 2 weeks. Assume CSL of 90%. Repeat problem 2.

21





Assume that 1 year = 12 months = 48 weeks Monthly demand is normally distributed with mean 85 units and

standard deviation 26 units. The replenishment lead is 6 weeks. Length of review period is 1 week (say every Saturday)

1. Given 90% CSL, find safety stock and OUTL. [Ans OUTL=193]

2. The company wants to maintain 90% CSL. a) What is the average inventory level?

[B + SS + cycle stk = 1.61 + 44.25 + 21.25/2= 56.485]

b) What is the observed fill rate? [Ans: 1-1.61/21.25 = 92.42%]

c) What is the average annual cost?

OF = 48*P(𝐷𝑅 > 0) = 48*(1-P(𝐷𝑅 ≤ 0)) = 48*[1-Φ(0−85∗0.25

26 0.25)] =48*(1-0.051065) = 45.55.

Avg setup cost per yr = 500*OF = 500*(45.55).

Avg holding cost per yr = 87.48*(avg inv) = 87.48*56.485

22





Assume that 1 year = 12 months = 48 weeks Monthly demand is normally distributed with mean 85 units and

standard deviation 26 units. The replenishment lead is 6 weeks. Length of review period is 1 week (say every Saturday)

Lead Time, L

Review Period, R

L+R

Demand in one period,

𝐷1

𝐷𝐿

𝐷𝑅

𝐷𝐿+𝑅

Order-up-to level (OUTL),

S

CDF

Loss Fn

E[order size]

Pipeline inv

E[Inv]

In-stock probability

Fill rate

E[backorder]

23

Safety stock

24

Four main components are 1. Capital cost or opportunity

cost (return that company could make on money tied up in inventory)

2. Inventory service cost (e.g., insurances and taxes)

3. Storage cost (warehousing space-related costs that change with level of inventory)

4. Inventory risk or shrinkage cost (e.g., obsolescence, damage)

Suppose that

Cost of capital 28%

Cost of storage 6%

Taxes & insurance 2%

Breakage & spoilage 1% Total percentage

37% An item valued at 180 THB would

have an annual holding cost of ℎ = 𝑐 ∗ 𝑟 = (180)(.37) = 66.6THB/unit/year

The holding cost (a.k.a. carrying cost or inventory cost) is the sum of all costs that are proportional to the amount of inventory physically on hand at any point in time.

• Suppose that we pay for the items upfront and take ownership of them as soon as the order is placed, we should also consider the cost of the 'pipeline inventory.'

• Holding pipeline inventory is a little cheaper than holding inventory on hand. • Cost of carrying inventory on hand includes both the cost of money (which is 28%

in the above example) and the cost of storing and managing the inventory in the DC (which accounts for the rest).

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Trade-offs

Everything else being equal:

• The higher the service level, the higher the inventory level.

• For the same inventory level, the longer the lead time to the facility, the lower the level of service provided by the facility.

• The lower the inventory level, the higher the impact of a unit of inventory on service level and hence on expected profit

What is the appropriate level of service?

May be determined by the downstream customer Retailer may require the supplier, to maintain a

specific service level

Supplier will use that target to manage its own inventory

Facility may have flexibility to choose appropriate level of service

More inventory is needed as demand uncertainty increases for any fixed fill rate.

The required inventory is more sensitive to the fill rate level as demand uncertainty increases

The tradeoff between inventory and fill rate with Normally distributed demand

and a mean of 100. The curves differ in the standard deviation of demand: 60,

50, 40, 30, 20, 10 from top to bottom.

26

Reducing the lead time reduces expected inventory, especially as the target fill rate increases

0

100

200

300

400

500

600

0 5 10 15 20Lead time

Exp

ecte

d i

nv

ento

ry

The impact of lead time on expected inventory for four fill rate targets,

99.9%, 99.5%, 99.0% and 98%, top curve to bottom curve respectively.

Demand in one period is Normally distributed with mean 100 and

standard deviation 60.

27

Reducing the lead time reduces expected inventory and pipeline inventory

The impact on pipeline inventory can be even more dramatic that the impact on expected inventory0

500

1000

1500

2000

2500

3000

0 5 10 15 20

Lead time

Invento

ry

Expected inventory (diamonds) and total inventory (squares), which is expected

inventory plus pipeline inventory, with a 99.9% fill rate requirement and demand

in one period is Normally distributed with mean 100 and standard deviation 60

28

Wal-Mart has consistently improved its annual inventory turns (approximately) the last two decades.

While a number of a factors could explain this dramatic improvement, reductions in its lead time is surely a significant factor.

These reductions were achieved through numerous initiatives. To improve the lead time Wal-Mart receives from its suppliers to its DCs, Wal-Mart build

electronic linkages with its suppliers. ▪ These linkages ensure that no time is wasted in order transmission and order processing. ▪ Furthermore, they allow Wal-Mart to share demand data with suppliers so that suppliers can ensure

they have enough capacity to meet Wal-Mart’s needs on a timely basis. (Lead times can be quite long if a supplier runs out of critical components or if the supplier runs out of capacity.)

Next, Wal-Mart designed its DCs and logistics so that inventory spends very little time in the DCs. ▪ For example, a popular product such as Crest toothpaste generally spends less than eight hours in a

Wal-Mart distribution center▪ Through a process called cross-docking, inventory is moved from in-bound trucks directly to out-bound

trucks, that is, it is never actually put on a shelf in the warehouse (Nelson 1999).

• Finally, via computerized replenishment and control of its own delivery fleet of vehicles, Wal-Mart’s lead time from its DCs to its stores is as fast as it can be.

• As a result of the combined impact of these initiatives, Wal-Mart is able to sell much of its inventory even before it must pay for that inventory, a rather enviable situation for any retailer.

29

1. Annual average setup cost (order freq)*(setup cost) = (OF)*(K) OF units/year = (annual demand rate)/E[order size] = d/E[order size] Setup cost = K THB/order

▪ Fixed ordering cost + (TL transportation cost)

2. Annual average holding cost (unit holding cost)*(avg on-hand inv) Unit holding cost, h THB/unit/year

▪ h = (unit cost)*(interest rate) = c*r ▪ unit cost = (variable ordering cost) + (LTL transportation cost)

Avg on-hand inv = (Cycle stock) + (SS + E[backorder]) Avg in-transit (pipeline) inventory

3. Annual average backorder cost4. Annual acquisition cost

No discount: (unit cost)*(annual demand rate) = c*d All-unit discount; unit cost depends on order quantity

30

• 𝒃1cost per stockout occasion (THB/occasion)

• Annual avg backorder cost = 𝑏1(order freq)*(stockout prob)

𝑏1THB

occasion* OF

cycle

year* stockout prob

occasion

cycle

• Recall E[Backorder at end of cycle] = 𝜎𝑒𝐿(𝑧) := ത𝐵• 𝒃2cost per unit short (THB/unit)

• Annual avg backorder cost = 𝑏2(order freq)* ത𝐵

𝑏2THB

unit* OF

cycle

year* ത𝐵

unit

cycle

• 𝒃3cost per unit short per yr (THB/unit/yr)

31

Average yearly demand is d = 200 units/year Unit holding cost = (24%)(2) = 0.48

THB/unit/year Cost per stockout occasion 𝑏1= 300 THB Demand during lead time 𝜇𝐿= 50 units, 𝜎𝐿=

21 units The order quantity is given as Q = 129 units

32

Yearly average cost Yearly average setup cost = K*OF = K*(d/Q) Yearly backorder cost = 𝑏1*OF*(stockout prob) = 𝑏1d/Q P(𝐷𝐿 > 𝑠) Yearly holding cost = h*(Q/2 + SS) = h*(Q/2 + (s-𝜇𝐿))

We want to choose ROP (s) to minimize yearly average cost Using Excel Solver (GRGNonlinear), we find that the optimal ROP is

𝑠∗ = 100.69 ≈ 101

ROP = s 100.69 units

b1 = cost per stockout occasion 300.00 THB/occasion

c = unit cost 2.00 THB/unit

h = unit holding cost 0.48 THB/unit/year

d = annual demand rate 200.00 unit/year

Q = order quantity 129.00 unit

OF = d/Q 1.55 times/year

muL 50.00 units

sigmaL 21.00 units

stockout prob 0.0079

annual avg backorder cost 3.67 THB/year

annual avg holding cost 55.29 THB/year

annual cost 58.96 THB/year

Average yearly demand is d = 200 units/year Unit holding cost = (24%)(2) = 0.48

THB/unit/year Cost per stockout occasion 𝑏1= 300 THB Demand during lead time 𝜇𝐿= 50 units, 𝜎𝐿=

21 units The order quantity is given as Q = 129 units

33

Yearly average cost Yearly average setup cost = K*OF = K*(d/Q) Yearly backorder cost = 𝑏1*OF*(stockout prob) = 𝑏1d/Q P(𝐷𝐿 > 𝑠) Yearly holding cost = h*(Q/2 + SS) = h*(Q/2 + (s-𝜇𝐿))

We want to choose ROP (s) to minimize yearly average cost Using Excel Solver (GRGNonlinear), we find that the optimal ROP is 𝑠∗ = 100.69 ≈ 101 Given that the setup cost is K = 120 THB/order and that the required CSL = 99.9%.

Determine the optimal pair of (s,Q).

ROP = s 100.69 units

b1 = cost per stockout occasion 300.00 THB/occasion

c = unit cost 2.00 THB/unit

h = unit holding cost 0.48 THB/unit/year

d = annual demand rate 200.00 unit/year

Q = order quantity 129.00 unit

OF = d/Q 1.55 times/year

muL 50.00 units

sigmaL 21.00 units

stockout prob 0.0079

annual avg backorder cost 3.67 THB/year

annual avg holding cost 55.29 THB/year

annual cost 58.96 THB/year

• Example CSCMP: A regional retailer, Value Dime and Five (VDF) has one DC that serves 500 stores. It only sells one SKU of toilet paper. It replenishes stores in case pack quantities, and each case contains 80 rolls.

• VDF only buys it by the truckload, which holds Q = 560 cases and pays 40 USD/case.• The transportation cost per truckload is 400 USD. All other costs of ordering associated with

purchasing, accounts payable, receiving about 50 USD/order.

Fixed order cost K = 400+50 = 450 USD/order• Demand during LT~normal mean of 𝝁𝑳=80 cases and a standard deviation of 𝝈𝑳=30 cases.

Lead time is 1 day.

34

Lead time 1 day

muL 80 units

sigmaL 30 units

fixed ordering cost 50 USD/order

c = unit cost 40 THB/case

holding cost factor for inv 0.25 per year

holding cost factor for in-transit 0.23 per year

backorder cost per unit short 5 USD/unit

TL cost 400 USD/truckload

1 TL = 560 case

Q 560 case

ROP 100 case

• Inventory carrying cost rate 25% /yr. In-transit inventory carrying cost rate 23% /yr

Unit carrying cost = (0.25)(40) = 10 USD/case/yr

Unit in-transit carrying cost = (0.23)(40) = 9.2 USD/case/yr

• The back-order cost is about 𝒃𝟐=5 USD/case. • VDF open every day of the yr; i.e., # of days in 1

yr=365. Average daily demand for DC is 80 cases/day. Average yearly demand d=(80)(365)=29200 cases/yr

Lead time 1 day

muL 80 units

sigmaL 30 units

fixed ordering cost 50 USD/order

c = unit cost 40 THB/case

holding cost factor for inv 0.25 per year

holding cost factor for in-transit 0.23 per year

backorder cost per unit short 5 USD/unit

TL cost 400 USD/truckload

1 TL = 560 case

Q 560 case

ROP 100 case

35

d = annual demand rate 29,200.00 case/year

OF = d/Q 52.14

setup cost = TL cost + fixed ordering cost 450.00 THB/time

annual ordering and transportation cost 23,464.29 THB/year

cycle stock 280.00

z = (ROP-muL)/sigmaL 0.67

L(z) 0.15

E[backorder per cycle] = sigmaL*L(z) := B 4.53 unit

annual avg backorder cost = b2*OF*B 1,181.97 THB/year

SS = ROP-muL 20 units

on-hand inventory 300.00 units

annual avg holding cost for on-hand 3000 THB/year

in-transit stock = muL 80 units

annual avg holding cost for in-transit 736 THB/year

annual avg cost 28,382.26 THB/year

# of truckload order quantity annual avg cost

31,456.18

1 560 28,382.26

2 1120 29,287.70

3 1680 31,456.18

4 2240 33,940.42

Input parameters Cost given Q = 560, ROP = 100

What-If using 1, 2, 3, 4 truckloads?

PERIODIC REVIEW: CONTROLLING

ORDERING COSTS

Suppose that the review period is four weeks, R=4 weeks.

• Effective demand, demand during L+R =8+4 =12 weeks, has mean 𝜇𝐿+𝑅 = 𝜇1 𝐿 + 𝑅 =100(12)=1200 and standard deviation 𝜎𝐿+𝑅 = 𝜎1 12=259.81

• The order frequency 1−𝑃(𝐷𝑅≤0)

𝑅=

1−Φ((0−100∗4)/(75∗ 4)

4=0.2490 /week or OF = 52(0.2490) = 12.95 /year.

• The annual fixed ordering cost K*(OF) = 275(12.95)=3561 THB/yr.

• The expected order size is ത𝑄 = 𝐸 𝐷𝑅 = 𝜇1𝑅=100(4)=400, and the cycle stock is ത𝑄/2=400/2 = 200.

• The desired in-stock probability is 99.25%. Safety factor z=Φ−1(0.9925)=2.43. SS =2.43𝜎𝐿+𝑅 =2.43(259.81) = 632

• OULT =𝜇𝐿+𝑅+SS =1200+ 632=1832.

• E[backorder] ത𝐵 = 𝜎𝐿+𝑅L(z)= 259.81L(2.43) = 259.81(0.002484)=0.6454

• E[ending inv] ҧ𝐼=SS+ ത𝐵=632+0.6454=632.6454

• E[On-hand inv] = (cycle stock) + E[ending inv] = 200 + 632 + 0.6454 =832.6454.

• The annual holding cost h*E[On-hand inv] = 12.5(832.6454)=10408.

• The expected total cost = K*(OF) + h*E[On-hand inv] = 3561+ 10408 = 13,969.37

36

Wkly demand

AVG 100

fixed ordering cost 275 STD 75

annual holding cost 12.5 Lead time,L 8

# of weeks in one yr 52

Review period 1 2 4 8

Target in-stock prob 0.9925 0.9925 0.9925 0.9925

OUTL 1832 1577 1832 2330

cycle stock 200 100 200 400

safety stock 632 577 632 730

Avg backorder 0.645 0.589 0.645 0.745

Avg ending inv 633 578 633 731

Avg on-hand inv 833 678 833 1131

order freq (times/yr) 12.95 25.23 12.95 6.50

annual fixedordering cost 3,561 6,938 3,561 1,787

inventory cost 10,408 8,470 10,408 14,134

total cost 13,969 15,408 13,969 15,922

Input parameters

37

• Our best option is to set the period

length to four weeks.

• A shorter period length results in

too many orders so the extra

ordering costs dominate the

reduced holding costs.

• A longer period suffers from too

much inventory.

Review period 1 2 3 4 5 6 7 8

Target in-stock prob 0.9925 0.9925 0.9925 0.9925 0.9925 0.9925 0.9925 0.9925

OUTL 1448 1577 1706 1832 1958 2083 2207 2330

cycle stock 50 100 150 200 250 300 350 400

safety stock 548 577 606 632 658 683 707 730

Avg backorder 0.559 0.589 0.618 0.645 0.672 0.697 0.722 0.745

Avg ending inv 549 578 607 633 659 684 708 731

Avg on-hand inv 599 678 757 833 909 984 1058 1131

order freq (times/yr) 47.26 25.23 17.15 12.95 10.39 8.66 7.43 6.50

annual fixedordering cost 12,996 6,938 4,717 3,561 2,856 2,382 2,042 1,787

inventory cost 7,482 8,470 9,458 10,408 11,358 12,296 13,222 14,134

total cost 20,478 15,408 14,175 13,969 14,214 14,678 15,264 15,922


ORDERING COSTS


ORDERING COSTS

• Although this analysis has been done in the context of the OUT model, it may very well remind you of another model, the EOQ model.

• Recall that in the EOQ model, there is a fixed cost per order/batch K = 275 THB, a holding cost per unit per unit of time h = 12.5 THB/unit/yr, and demand occurs at a constant flow rate d.

• In this case, yearly demand d = (52)(100)=5200 unit/yr.

• The EOQ is 2𝐾𝑑

ℎ=

2(275)(5200)

12.5=478. This implies a cycle time of EOQ/d = 478/5200

yr = (478/5200)*52 = 4.78 weeks: An order should be submitted every 4.78 weeks.

• The key difference between our model and the EOQ model is that here we have random demand whereas the EOQ model assumes demand occurs at a constant rate.

• Even though the OUT model and the EOQ models are different, the EOQ model gives a very good recommendation for the period length.

38

Review period 1 2 3 4 5 6 7 8

Target in-stock prob 0.9925 0.9925 0.9925 0.9925 0.9925 0.9925 0.9925 0.9925

OUTL 1448 1577 1706 1832 1958 2083 2207 2330

cycle stock 50 100 150 200 250 300 350 400

safety stock 548 577 606 632 658 683 707 730

Avg backorder 0.559 0.589 0.618 0.645 0.672 0.697 0.722 0.745

Avg ending inv 549 578 607 633 659 684 708 731

Avg on-hand inv 599 678 757 833 909 984 1058 1131

order freq (times/yr) 47.26 25.23 17.15 12.95 10.39 8.66 7.43 6.50

annual fixedordering cost 12,996 6,938 4,717 3,561 2,856 2,382 2,042 1,787

inventory cost 7,482 8,470 9,458 10,408 11,358 12,296 13,222 14,134

total cost 20,478 15,408 14,175 13,969 14,214 14,678 15,264 15,922

Wkly demand

AVG 100

fixed ordering cost 275 STD 75

annual holding cost 12.5 Lead time,L 8

# of weeks in one yr 52

Input parameters

PERIODIC REVIEW &

JOINT REPLENISHMENT

• EOQ formula gives us an easy way to check if our period length is reasonable.

• One advantage of this approach is that we submit orders on a regular schedule.

• This is a useful feature if we need to coordinate the orders across multiple items.

• For example, since we incur a fixed cost per truck shipment, we generally deliver many different products on each truck, because no single product’s demand is large enough to fill a truck.

• In that situation, it is quite useful to order items at the same time so that the truck can be loaded quickly and we can ensure a reasonably full shipment (given that there is a fixed cost per shipment, it makes sense to utilize the cargo capacity as much as possible).

• Therefore, we need only ensure that the order times of different products align.

39

CONTROLLING

ORDERING COSTS

• Instead of using fixed order intervals, as in the OUT model, we could control ordering costs by imposing a minimum order quantity.

• For example, we could wait for Q units of demand to occur and then order exactly Q units.

• With such a policy, we would order on average every Q/d units of time, but due to randomness in demand, the time between orders would vary.

• Not surprisingly, the EOQ quantity provides an excellent recommendation for that minimum order quantity.

• Important insight is that it is possible to control ordering costs by

1) restricting to a periodic schedule of order, or

2) restricting to a fixed order quantity.

• With the first option, there is little variability in the timing of orders, which facilitates the coordination of orders across multiple items, but the order quantities are variable (which may increase handling costs).

• With the second option, the order quantities are not variable (we always order Q), but the timing of those orders varies. 4

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