Strategic Inventory Deployment forMulti-channel Businesses

Bhaskar Ramasubramanian

Electrical and Computer EngineeringUMD, College Park

December 10, 2012

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References

Yao, D. Q., Yue, X., Mukhopadhyay, S. K., and Wang, Z. (2009),Strategic inventory deployment for retail and e-tail stores,Omega, 37(3), 646− 658.

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Preliminaries

Click and mortar companies.Most e-business failures are operations related - late delivery/stock out common during busy periods.Formulating optimal inventory decisions for retail and e-tail armsof the business for various strategies.

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Business Strategies Considered - CentralizedInventory Strategy

Manufacturer owns both retail and e-tail channels.Manufacturer has full control of inventory decisions.Only one centralized decision maker.

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Business Strategies Considered - StackelbergInventory Strategy

Manufacturer owns e-tail store, while retailer controls retail store.Manufacturer has full control of the e-tail inventory level andwholesale price to the retailer.Retailer accordingly makes inventory decisions for the retail arm.Leader - follower game.

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Business Strategies Considered - OutsourcingInventory Strategy

Manufacturer owns e-tail store, while retailer controls retail store.E-tail channel outsourced to third party logistic provider.Manufacturer makes inventory decision on the e-tail channelmanaged by the third party.

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The Model

Supply chain with retail and e-tail channels selling goods withshort life cycle.Customers can make purchases at a retail store or place anorder online.Total demand is a random variable, and a certain fraction ofcustomers prefer using the retail channel.Model decides the inventory level that maximizes profits for asingle period, under probabilistic demand.If a customer’s demand could not be met on the selectedchannel, the demand is lost.If a product is out of stock in a channel, there is no transshipmentfrom the other channel.

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The Model - Notation

i = 1 - e-tail channel; i = 2 - retail channel.ci - unit cost for channel i .di - demand for channel i ; d = d1 + d2 - total demand.f (d) - probability density function of d .fi (di ) - probability density function of channel i .gi - unit shortage cost for channel i .pi - unit sale price for channel i .Qi - inventory quantity for channel i .si - salvage value per unit of the product for channel i .c̃i - unit cost of outsourcing for the e-tail channel.

Assume the demand is split from a certain distribution with thedemand for the online channel d1 = s(d) and that for the retailchannel being d2 = d − s(d).

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The Model - Assumptions

New customer more likely to use the e-tail rather than the retailchannel, i.e. s′(d) > 0 and s′′(d) > 0.Larger number of customers use the retail channel.

Total demand, d ∼ Unif [0,1]. Thus, f (d) = 1.s(d) = θd2 and 0 < θ < 0.5.

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Lemma

If f (d) = 1, then f1(d1) = 1/2√θd1 and f2(d2) = 1/

√1− 4θd2.

Proof : fY (y) = fX |g−1(y)|| dg−1(y)dy |.

Here, d1 = θd2 ⇒ d =√

d1θ . Thus f1(d1) = 1/2

√θd1.

Similarly, d2 = d − θd2 ⇒ d = 1±√

1−4θd22θ .

θ < 0.5 means that one value of d > 1, which is not feasible.Therefore, f2(d2) = 1/

√1− 4θd2.

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Centralized Inventory Strategy

Total Expected Profit for Two Channels

E[π] = −c1Q1 − c2Q2

+[

∫ Q1

0[p1d1 + s1(Q1 − d1)] +

∫ θ

Q1

[p1Q1 − g1(d1 −Q1)]]f1(d1)d(d1)

+[

∫ Q2

0[p2d2 + s2(Q2 − d2)] +

∫ 1−θ

Q2

[p2Q2 − g2(d2 −Q2)]]f2(d2)d(d2)

(1)

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Centralized Inventory Strategy

Proposition

The optimal stocking size for online and retail stores are as follows :

Q∗1 = θ[

p1+g1−c1p1+g1−s1

]2(2)

Q∗2 =[

p2+g2−c2p2+g2−s2

]− θ

[p2+g2−c2p2+g2−s2

]2(3)

Proof : Set ∂E[π]/∂Q1 and ∂E[π]/∂Q2 to 0 and solve for Q1 and Q2,by using the values of f1(d1) and f2(d2) from previous Lemma.Also, ∂2E[π]/∂Q2

1 < 0 and ∂2E[π]/∂Q22 < 0 and

∂2E[π]/∂Q1∂Q2 = ∂2E[π]/∂Q2∂Q1 = 0, which means the Hessianmatrix is negative definite.

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Stackelberg (leader-follower) inventory strategy

The two channels are de facto competitors. Thus, they makedecisions independently keeping their own interests in mind.The manufacturer is the Stackelberg inventory leader while theretailer is the follower.The manufacturer makes decisions on the online inventory levelQ1 and the wholesale price w offered to the retailer, in order tomaximize his expected profit, given the expected responsefunction of the retailer.In response to the manufacturer’s wholesale price w , the retailerplaces an order to the manufacturer to stock an inventory levelQ2 so as to maximize her own expected profit.This requires the solution to two problems.

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Stackelberg Strategy - Retailer’s Problem

Retailer’s Expected Profit, given w

E[π2|w ] = −wQ2

+[

∫ Q2

0[p2d2 + s2(Q2 − d2)] +

∫ 1−θ

Q2

[p2Q2 − g2(d2 −Q2)]]f (d2)d(d2)

(4)

where 1− θ is the retailer’s maximum demand.

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Stackelberg Strategy - Retailer’s Problem

Proposition

The retailer’s best inventory response function is :

Qs∗2 =

[p2+g2−wp2+g2−s2

]− θ

[p2+g2−wp2+g2−s2

]2(5)

Proof : Similar to previous proposition.

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Stackelberg Strategy - Manufacturer’s Problem

Manufacturer’s Objective

maxQ1,w

E[π1] = −c1Q1 + (w − c2)Q2

+[

∫ Q1

0[p1d1 + s1(Q1 − d1)] +

∫ θ

Q1

[p1Q1 − g1(d1 −Q1)]]f (d1)d(d1)

(6)

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Stackelberg Strategy - Manufacturer’s Problem

Proposition

The manufacturer’s optimal decisions are :

w∗ =13θ[−g2 − p2 + s2 + c2θ + 2(g2 + p2)θ +

√∆]

(7)

Qs∗1 = θ

[p1+g1−c1p1+g1−s1

]2(8)

where∆ = (g2 + p2− s2)2− (g2 + p2− c2)(p2 + g2− s2)θ+ (p2 + g2− c2)2θ2.

Proof : Refer paper

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Stackelberg Strategy - Manufacturer’s Problem

Corollary

∂w∗/∂θ > 0

Proof :Refer paper.

If the manufacturer enjoys a larger θ, he can increase hiswholesale price.

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Stackelberg Strategy - Manufacturer’s Problem

More customers switch to the e-tail mode⇒ it would be optimalfor the manufacturer to increase wholesale price.This, in turn, would discourage the retailer to stock moreinventory from the manufacturer.The manufacturer’s expected profit is thus :

E[πs∗1 ] = (w∗ − c2)Qs∗

2 +13

(p1 + g1 − c1)Qs∗1 −

13

g1θ (9)

Also, Qs∗1 = Q∗1 , but Qs∗

2 ≤ Q∗2 , where Q∗1 and Q∗2 are the optimalinventory levels for the centralized strategy.

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Outsourcing Inventory Strategy

Assume that the new unit cost for the e-tail channel is c̃1 ≤ c1.The manufacturer pays the third party a transaction fee φ foreach unit sold via the e-tail channel.The optimal inventory level for the retailer and the optimalwholesale price is the same as in Propositions above for theStackelberg strategy.

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Outsourcing Inventory Strategy

Expected Profit for Manufacturer

maxQ1,w

E[π1] = −c̃1Q1 + (w − c2)Q2

+[

∫ Q1

0[(p1 − φ)d1 + s1(Q1 − d1)]

+

∫ θ

Q1

[(p1 − φ)Q1 − g1(d1 −Q1)]]f (d1)d(d1) (10)

φ is the reduction in the e-tail price due to the reduced unit costin this case.

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Outsourcing Inventory Strategy

Proposition

The optimal inventory size for an e-tail store following an outsourcingstrategy is :

Qout∗1 = θ

[p1+g1−c̃1−φp1+g1−s1−φ

]2(11)

Corollary

If :

c1 − c̃1

φ≥ c1 − s1

g1 + p1 − s1

holds, then Qout∗1 ≥ Qs∗

1 . Else, Qout∗1 < Qs∗

1 .

c1−c̃1φ is called the outsourcing efficiency.

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Numerical Experiments

The manufacturer’s expected profit in the outsourcing case is :

E[πout∗1 ] = (w∗ − c2)Qout∗

2 +13

(p1 + g1 − c̃1 − φ)Qout∗1 − 1

3g1θ

When there is no cost savings by outsourcing, the profit is alwayslower than Stackelberg - because of the fee φ paid to the thirdparty per unit sold.A low value of φ yields more profit in the outsourcing case thanthe Stackelberg case.Since Stackelberg and outsourcing strategies share the samewholesale price w∗ and inventory levels Qout∗

2 = Qs∗2 , the

retailer’s expected profit in either case is the same, i.e.E[πout∗

2 ] = E[πs∗2 ].

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Numerical Experiments

Centralized strategy keeps more inventory than in the other twocases.As the market share for the e-tail strategy goes up, i.e. as θincreases, the inventory levels for the centralized strategy isalmost constant, while that for the other two increase.Impact on production levels - stable for centralized strategy,therefore capacity planning easier.

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Graphs

Figure: Stackelberg vs. Outsourcing I

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Graphs

Figure: Stackelberg vs. Outsourcing II

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Graphs

Figure: Stackelberg vs. Outsourcing III

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Concluding Remarks

Profit maximization model for each strategy presented.As the share of e-channel demand increases, the inventorylevels increases in the e-channel and decreases in the retailchannel in different proportions.Even if the manufacturer pays the third party a fee for outsourcingthe e-channel inventory, he can still receive a higher profit.Total demand was uniformly distributed - approach can beextended to other distributions.Customer switching between channels not considered.

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Thank You

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