optimal inventory control with retail pre-packs · pdf fileproblem motivation optimal policies...

Problem MotivationOptimal policies

Computational Investigation

Optimal Inventory Control with RetailPre-Packs

Long Gao

joint work with Michael Freimer and Doug Thomas

Smeal College of BusinessThe Pennsylvania State University

March 6, 2007

Long Gao Optimal Inventory Control with Retail Pre-Packs

Outline

1 Problem Motivation

2 Optimal policiesSingle SKU Pre-packMultiple SKU pre-pack

3 Computational InvestigationImpact of Problem ParametersComparison of policies

Outline

Problem motivation

Two Forms of Packaging

ContractManufacturer

DistributionCenter

RetailStore

loose items

pre-packs

U.S.-based apparel retailerArkansas-based consumer goods retailer

DistributionCenter

RetailStore

loose items

pre-packs

DistributionCenter

RetailStore

loose items

pre-packs

When To Use Pre-Packs?

Significant savings in handlingand delivery costsReduces flexibility in ordering,increasing holding and shortagecostsBasic Tradeoff:handling savings (pre-pack) v.s.ordering flexibility (loose)

SavingsThe Limitedbrands reported 4% distribution costs savings in2006.

Current Practice at The Limitedbrands

Configure pre-packs with lead time 13 weeksBlack Pique Polo Pre-Pack: [1-2-2-1]

Set inventory targets, ignoring the prep-packconfigurationAllocate prepacks and loose to stores

HoweverThe pre-pack configuration is not necessarily optimal.The inventory policy is not optimal.

Inventory Policies: (R, nQ) and Base stock

T im e

y Q : P rep ack Size

R: T arget Level

(R,n Q ) P o licy

QuestionWhat if we can order both pre-pack and loose?

T im e

y Q : P rep ack SizeS: T arget Level

Base Sto ck P o licy

(R,n Q ) P o licy

T im e

y Q : P rep ack SizeS: T arget Level

Base Sto ck P o licy

(R,n Q ) P o licy

Research Questions

Policy Characterization: What is the optimal ordering policyfor the retail store if both pre-pack and loose can beordered?

Policy Comparison: How does the optimal policy compareto other simpler policies, i.e., base stock and (R, nQ)?

Pre-Pack Design: What should go in the pre-pack?

Research Questions

Relevant Literature and Contributions

Optimal inventory policies(R,nQ) Policy

Hadley & Whitin (1965)Zheng & Chen (1992)

Henig et. al. (1997)Parkinson & McCormick (2005)

Pre-pack designAgrawal & Smith (2003)

Our ContributionsTheoretical: characterization of the whole class ofinventory polices with both pre-pack and loose optionsPractical: recommendations for inventory policy selectionand pre-pack design

Single SKU Pre-packMultiple SKU pre-pack

Outline

System Assumptions

Periodic review inventory system (1-3 times per week)

Zero ordering cost (fixed transportation cost)

Zero lead time (next day delivery)

Stationary demand (short term, operational level)

System Assumptions

Some notation

Q = size of the pre-packδ = per-unit handling penalty for ordering looseh = per unit holding costb = per unit shortage costξ = random demand

x = initial inventory

y0 = number of loose units ordered, y0 < Q

y1 = number of pre-packs ordered

I = target inventory position, I = x + y0 + y1Q

Some notation

Sequence of events

Start with initial inventory x

Place the order for y0 + y1Qbring the inventory position to I = x + y0 + y1Qincurring handling cost δy0 for ordering loose

Receive the order

Observe the demand ξ

Fulfill the demand

Ending inventory is (I − ξ)

incurring holding/shortage cost h(I − ξ)+ + b(ξ − I)+

Sequence of events

Receive the order

Fulfill the demand

Sequence of events

Receive the order

Fulfill the demand

Sequence of events

Receive the order

Fulfill the demand

Sequence of events

Receive the order

Fulfill the demand

Sequence of events

Receive the order

Fulfill the demand

Sequence of events

Receive the order

Fulfill the demand

Sequence of events

Receive the order

Fulfill the demand

Sequence of events

Receive the order

Fulfill the demand

Sequence of events

Receive the order

Fulfill the demand

Problem formulation

Obtain the optimal policy π∗ by solving a Markov DecisionProcess (MDP):

V(x) = minπ

δy0︸︷︷︸

handling cost

+ Eξ[h(I − ξ)+ + b(ξ − I)+]︸︷︷︸inventory cost

+βEξV(I − ξ)︸︷︷︸future cost

Cost of pre-pack only:

G(I) := Eξ[h(I − ξ)+ + b(ξ − I)+] + βEξV(I − ξ)

Minimum pre-packs:

I(x) := maxx + y1Q : x + y1Q ≤ S

Cost of ordering both pre-pack and loose:

H(x, I) := δ(I − I(x))+ + G(I)

Problem formulation

V(x) = minπ

δy0︸︷︷︸

handling cost

Minimum pre-packs:

I(x) := maxx + y1Q : x + y1Q ≤ S

H(x, I) := δ(I − I(x))+ + G(I)

Problem formulation

V(x) = minπ

δy0︸︷︷︸

handling cost

Minimum pre-packs:

I(x) := maxx + y1Q : x + y1Q ≤ S

H(x, I) := δ(I − I(x))+ + G(I)

Problem formulation

V(x) = minπ

δy0︸︷︷︸

handling cost

Minimum pre-packs:

I(x) := maxx + y1Q : x + y1Q ≤ S

H(x, I) := δ(I − I(x))+ + G(I)

Optimal Policy

0 Inventory Position I(x)

SS−Q

G( I ): Prepack Only

Optimal Policy

SZS−Q

Ω1 Ω

H(x,I): Prepack & Loose

Optimal Policy

B+QSZBS−Q

Ω1 Ω

Optimal Policy

B+QSZBS−Q

Ω1 Ω

Optimal Policy

B+QSZBS−Q

Ω1 Ω

Optimal Policy

B+QSZBS−Q

Ω1 Ω

Optimal Policy

Condition (C):Loose target Z: ∃Z ∈ (S − Q, S) s.t. G′(Z) = −δ

Indifferent point B: ∃B ∈ (S − Q, Z] s.t. H(B, Z) = G(B + Q)

Optimal Policy:Under (C),

π∗(C) ≡

order y1Q to bring I = I(x) + Q, if I(x) ∈ Ω1

order y0 + y1Q to bring I = Z, if I(x) ∈ Ω2

order y1Q to bring I = I(x), if I(x) ∈ Ω3

Under (C)c, (R, nQ) policy is optimal, where R satisfiesG(R) = G(R + Q).If δ = 0, base stock policy is optimal.

Optimal Policy

π∗(C) ≡

Optimal Policy

π∗(C) ≡

Optimal Policy

π∗(C) ≡

Optimal Policy

π∗(C) ≡

Optimal Policy

B+QSZBS−Q

Ω1 Ω

Steady State Distribution

Theorem:Under (C), if demand follows a discrete uniform distribution on[0, nQ − 1], and the band policy [Z, B + Q] is implemented, thenthe steady-state inventory position has a semi-uniformdistribution:

Q , 1Q , . . . , 1

on Z, Z − 1, . . . , B + Q

Application:For large scale problems, the theorem is useful to developefficient approximations.

Steady State Distribution

Theorem:Under (C), if demand follows a discrete uniform distribution on[0, nQ − 1], and the band policy [Z, B + Q] is implemented, thenthe steady-state inventory position has a semi-uniformdistribution:

Q , 1Q , . . . , 1

on Z, Z − 1, . . . , B + Q

Application:For large scale problems, the theorem is useful to developefficient approximations.

Steady State Distribution: Proof

For each initial inventory i, classify demand values (ξ ∼ f )into two sets: order only prepack (∪nEin) and order both(∪nEin)

c, where Ein = i− E + nQ, E = Z, Z + 1, . . . , B + Q.Transition probability:

if j 6= Z,pij = P ξ ∈ ∪nEin : ξ = nQ + i − j

=∞∑

f (nQ + i − j)

if j = Z,

pij = P ξ ∈ (∪nEin)c + P ξ ∈ ∪nEin : ξ = i − Z + nQ

= 1 −B+Q∑

∞∑n=0

f (nQ + i − l).

=∞∑

f (nQ + i − j)

if j = Z,

= 1 −B+Q∑

∞∑n=0

f (nQ + i − l).

=∞∑

f (nQ + i − j)

if j = Z,

= 1 −B+Q∑

∞∑n=0

f (nQ + i − l).

=∞∑

f (nQ + i − j)

if j = Z,

= 1 −B+Q∑

∞∑n=0

f (nQ + i − l).

=∞∑

f (nQ + i − j)

if j = Z,

= 1 −B+Q∑

∞∑n=0

f (nQ + i − l).

=∞∑

f (nQ + i − j)

if j = Z,

= 1 −B+Q∑

∞∑n=0

f (nQ + i − l).

=∞∑

f (nQ + i − j)

if j = Z,

= 1 −B+Q∑

∞∑n=0

f (nQ + i − l).

=∞∑

f (nQ + i − j)

if j = Z,

= 1 −B+Q∑

∞∑n=0

f (nQ + i − l).

Transition matrix P exhibits certain periodic property:

P = [pij] =

dZ a−1 a−2 . . . aZ−(B+Q)

dZ+1 a0 a−1 . . . aZ+1−(B+Q)

dZ+2 a1 a0 . . . aZ+2−(B+Q)...

......

. . ....

dB+Q aB+Q−Z−1 aB+Q−Z−2 . . . a0

where ai =

∑∞n=0 f (nQ + i), di = 1 −

∑B+Ql=Z+1 ai−l.

If ξ ∼ U[0, nQ − 1], then ai = 1/Q, di = (Z − Q)/Q.Transition matrix P can be simplified as:

d a . . . ad a . . . a...

.... . .

...d a . . . a

(Z−B

Q , 1Q , . . . , 1

)satisfies

φ ·P = φ∑j∈E

φj = 1

For irreducible Markov Chain, the solution is unique.

d a . . . ad a . . . a...

.... . .

...d a . . . a

(Z−B

Q , 1Q , . . . , 1

)satisfies

φ ·P = φ∑j∈E

φj = 1

d a . . . ad a . . . a...

.... . .

...d a . . . a

(Z−B

Q , 1Q , . . . , 1

)satisfies

φ ·P = φ∑j∈E

φj = 1

d a . . . ad a . . . a...

.... . .

...d a . . . a

(Z−B

Q , 1Q , . . . , 1

)satisfies

φ ·P = φ∑j∈E

φj = 1

Multiple SKUs

Retailer may wish to put multiple SKUs in a pre-pack:Different sizesDifferent colorsMatching sets (e.g. hats, scarves)

Demand correlation plays a significant role

Multiple SKUs

Problem formulation:Pre-pack: Q = (Q1, Q2, . . . , Qn)

Loose units: y0 = (y10, y2

0, . . . , yn0)

Penalty cost: δ · y0 =∑n

i=1 δiyi0

As before, optimal policy π∗ is obtained by solving MDP:V(x) = minπδy0 + L(I) + βEV(I − ξ)

Optimal policy for two SKUs

−6 −4 −2 0 2 4 6 8 10

Impact of Problem ParametersComparison of policies

Outline

Computational Results

Impact on optimal policy of:per-unit handling penalty δpre-pack size Qdemand variability cvdemand correlation ρ (across multiple SKUs)

Comparison between:optimal policybase-stock policy(R, nQ) policy

Criteria:target inventory region and distributionpre-pack usagetotal cost

Scenario for comparison

Two-SKUs with symmetric pre-pack of sizeQ ∈ (3, 3), (4, 4), (5, 5), (6, 6)unit penalty cost δ ∈ 0.01, 0.5, 1, 5Demand: discretized Normal with µ = 5 andcv ∈ 0.1, 0.2, 0.3, 0.34Demand correlation ρ ∈ −0.6, 0, 0.6unit holding cost h = 1 and shortage cost b = 10

Base case: Q = (4, 4), δ = 0.5, cv = 0.3, ρ = 0

Impact of per-unit handling penalty δ

4 5 6 7 8 9 10 114

SKU1 Inventory

δ = 0.01

4 5 6 7 8 9 10 114

SKU1 Inventory

δ = 0.01

4 5 6 7 8 9 10 114

SKU1 Inventory

δ = 0.5

4 5 6 7 8 9 10 114

SKU1 Inventory

δ = 0.01

4 5 6 7 8 9 10 114

SKU1 Inventory

δ = 0.5

4 5 6 7 8 9 10 114

SKU1 Inventory

δ = 1

4 5 6 7 8 9 10 114

SKU1 Inventory

δ = 0.01

4 5 6 7 8 9 10 114

SKU1 Inventory

δ = 0.5

4 5 6 7 8 9 10 114

SKU1 Inventory

δ = 1

4 5 6 7 8 9 10 114

SKU1 Inventory

δ = 5

Table: Impact of per-unit handling penalty δ

δ Prepack Usage Total Cost

0.01 54.1% 5.180.50 80.8% 6.471.00 91.0% 7.005.00 94.2% 7.62

Target inventory region (after ordering) expands, avoidingexcess handling costsTotal cost and pre-pack usage increase as δ increasesBase stock policy is desirable for small handling cost δ

Pre-pack only policy is optimal for large handing savings δ

0.01 54.1% 5.180.50 80.8% 6.471.00 91.0% 7.005.00 94.2% 7.62

Impact of Pre-Pack Size Q

4 5 6 7 8 9 10 114

SKU1 Inventory

2)= (3,3)

4 5 6 7 8 9 10 114

SKU1 Inventory

2)= (3,3)

4 5 6 7 8 9 10 114

SKU1 Inventory

2)= (4,4)

4 5 6 7 8 9 10 114

SKU1 Inventory

2)= (3,3)

4 5 6 7 8 9 10 114

SKU1 Inventory

2)= (4,4)

4 5 6 7 8 9 10 114

SKU1 Inventory

2)= (5,5)

4 5 6 7 8 9 10 114

SKU1 Inventory

2)= (3,3)

4 5 6 7 8 9 10 114

SKU1 Inventory

2)= (4,4)

4 5 6 7 8 9 10 114

SKU1 Inventory

2)= (5,5)

4 5 6 7 8 9 10 114

SKU1 Inventory

2)= (6,6)

Table: Impact of Package Size Q

(Q1, Q2) Prepack Usage Total Cost

(3,3) 83.5% 6.29(4,4) 80.8% 6.47(5,5) 76.1% 6.61(6,6) 71.5% 7.04

Target region expands, due to less ordering flexibilityTotal cost increases, and pre-pack usage decreasesWith the same handling savings, ordering a larger pre-packis less attractive to the store

(3,3) 83.5% 6.29(4,4) 80.8% 6.47(5,5) 76.1% 6.61(6,6) 71.5% 7.04

Impact of demand variability CV

4 5 6 7 8 9 10 114

SKU1 Inventory

cv = 0.1

4 5 6 7 8 9 10 114

SKU1 Inventory

cv = 0.1

4 5 6 7 8 9 10 114

SKU1 Inventory

cv = 0.2

4 5 6 7 8 9 10 114

SKU1 Inventory

cv = 0.1

4 5 6 7 8 9 10 114

SKU1 Inventory

cv = 0.2

4 5 6 7 8 9 10 114

SKU1 Inventory

cv = 0.3

4 5 6 7 8 9 10 114

SKU1 Inventory

cv = 0.1

4 5 6 7 8 9 10 114

SKU1 Inventory

cv = 0.2

4 5 6 7 8 9 10 114

SKU1 Inventory

cv = 0.3

4 5 6 7 8 9 10 114

SKU1 Inventory

cv = 0.34

Table: Impact of demand variability CV

CV Prepack Usage Total Cost

0.10 88.2% 2.720.20 82.5% 4.440.30 80.8% 6.470.34 79.4% 7.06

More volatile demand requires higher inventory level, andresults in higher total costPre-pack usage decreases as demand variability increasesPre-pack is attractive for relative stable demand

0.10 88.2% 2.720.20 82.5% 4.440.30 80.8% 6.470.34 79.4% 7.06

Impact of demand correlation ρ

Table: Impact of demand correlation ρ

ρ Prepack Usage Total Cost

-0.6 78.5% 6.580.0 80.8% 6.470.6 82.7% 6.28

Total cost decreases as ρ increases.Prepack usage increases as ρ increases.

Impact of demand correlation ρ

Table: Impact of demand correlation ρ

ρ Prepack Usage Total Cost

-0.6 78.5% 6.580.0 80.8% 6.470.6 82.7% 6.28

Total cost decreases as ρ increases.Prepack usage increases as ρ increases.

Impact of ρ (demand correlation)

4 5 6 7 8 9 10 114

SKU1 Inventory

ρ = −0.6

4 5 6 7 8 9 10 114

SKU1 Inventory

ρ = −0.6

4 5 6 7 8 9 10 114

SKU1 Inventory

ρ = 0

4 5 6 7 8 9 10 114

SKU1 Inventory

ρ = −0.6

4 5 6 7 8 9 10 114

SKU1 Inventory

ρ = 0.6

4 5 6 7 8 9 10 114

SKU1 Inventory

ρ = 0

4 5 6 7 8 9 10 114

SKU1 Inventory

ρ = −0.6

4 5 6 7 8 9 10 114

SKU1 Inventory

ρ = 0.6

Same region, but differentdistributionOrdering pre-packs isordering along thediagonal, ρ > 0Recommendation: bundledifference size of the samestyle-color

Limited: rainbow-pack,different color

4 5 6 7 8 9 10 114

SKU1 Inventory

ρ = −0.6

4 5 6 7 8 9 10 114

SKU1 Inventory

ρ = 0.6

4 5 6 7 8 9 10 114

SKU1 Inventory

ρ = −0.6

4 5 6 7 8 9 10 114

SKU1 Inventory

ρ = 0.6

4 5 6 7 8 9 10 114

SKU1 Inventory

ρ = −0.6

4 5 6 7 8 9 10 114

SKU1 Inventory

ρ = 0.6

Relative Costs of Simple Policies

0 2 4 6 8 10 12 140

Prepack Size: Q

Base Stock Policy

(R,nQ) Policy

Ω2 Ω

∆TC%: percentagetotal cost differenceΩ1:(Big Stores)(R, nQ) dominatesΩ2: Both aresuboptimal. Thecost gap can be30% higher!Ω3: (Small Stores)Base stock policy ispreferable.

0 2 4 6 8 10 12 140

Prepack Size: Q

Base Stock Policy

(R,nQ) Policy

Ω2 Ω

0 2 4 6 8 10 12 140

Prepack Size: Q

Base Stock Policy

(R,nQ) Policy

Ω2 Ω

0 2 4 6 8 10 12 140

Prepack Size: Q

Base Stock Policy

(R,nQ) Policy

Ω2 Ω

0 0.2 0.4 0.6 0.8 1.0 1.20

Unit Handling Penalty: δ

Base Stock Policy(R,nQ) Policy

Value of Flexibility

Value of flexibility:21%!R1: Base stockpolicy is attractive.R2: (R, nQ) policy isbetter.

0 0.2 0.4 0.6 0.8 1.0 1.20

Conclusions

Summary:Characterized the forms of the optimal policy for the singleand multiple SKU casesInvestigated the sensitivity of the optimal policy to problemparametersProvided guidance for policy selection and pre-pack design

Future work:How should the pre-pack be designed?Common pre-packs for multiple retail locationsMultiple pre-packs

Conclusions

Other Applications of Our Model

Contract transportation:truckload and less thantruckload

Chemical suppliers:large and small ocean tanks

Other Applications of Our Model

Contract transportation:truckload and less thantruckload

Chemical suppliers:large and small ocean tanks

Questions?

optimal inventory control with retail pre-packs · pdf fileproblem motivation optimal policies...

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