Optimal Procurement, Pricing and Disposal Policies for ...

[Pages:6]Optimal Procurement, Pricing and Disposal Policies for Managing Rental Goods

Sarang Deo and Christopher S. Tang

UCLA Anderson School of Management

Video rental industry has grown signi cantly recently, especially since the introduction of the DVD format. DVD rental market in the U.S. alone accounted for $5 billion in year 2003 and is estimated to grow to $10 billion by year 2008 (based on Net ix annual report, ). Based on weekly rental data provided on , we notice that rental demand for each title decreases over time as the popularity of the movie declines and as the number of persons who have already viewed the movie increases. As rental demand drops below a certain level, the video rental retailer would need to dispose some inventory procured earlier and move the title from the `new releases' section to the `catalog' section. This need to dispose inventory is accentuated by the limited shelf space and the need to make space for newer titles. In the case of linear pricing contracts, where the retailer buys the stock outright, the responsibility to dispose excess inventory rests with the retailer. While doing so, the retailer faces important decisions like timing of disposal, quantity to be disposed and the selling price. Moreover, given the decreasing nature of the rental demand, the retailer also needs to decide the initial quantity to be ordered from the distributor so that he can satisfy the higher initial rental demand without incurring very high holding and procurement costs.

In this paper, we aim to derive the optimal stocking and disposal policy for a video rental retailer when the underlying rental demand is deterministic and decreasing over time. In practice, the demand might not be known to the retailer with certainty, but can be predicted fairly accurately by using box o ce collections and DVD sales. We rst develop a base model, where we assume that the disposal price is provided exogeneously to the retailer, that all the rental demand has to be met exactly in each period and that there is a xed cost per period associated with the disposal. We also discuss the endogeneous price case. We nd that when the demand for disposed quantity is linear in disposal price, the decision problem of the retailer can be modeled as a quadratic optimization problem which can be solved to optimality by dualization. Later we discuss various extensions to this model which incorporate following cases (i) multiple procurement opportunities (ii) contractual period of no sale (iii) backordering of unmet demand. We nd that the qualitative nature of the optimal policy is preserved in all of these cases but the actual computation and quantitative results vary in each. For brevity, only the salient result of the

rst extension is discussed. Details of all extensions can be found in Deo and Tang (2005)

Considerable amount of research has been done to analyze various contracting arrangements across the vertical tiers in the video rental industry. The reader is referred to Dana and Spier (2001) and Cachon and Lariviere (2005) for more details. Mortimer (2004) conducts an empirical analysis to examine the e ect of revenue sharing scheme on the retailer's pro t. Relatively less attention has been devoted to the nature of the competition among retailers in the rental business and even lesser to the operational

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decisions of a single retailer. Dana (2001) presents a strategic, single period model of competition among retailers in price and availability. Tang and Deo (2004) analyze the decisions of a single retailer facing uncertain rental demand and return process in the context of a multi-period model. They also analyze the duopoly model and establish conditions under which one retailer competes on price and the other competes on the rental duration. Another stream of literature closely related to our paper is the one addressing the question of optimal disposal policies. In this literature, the approach closest to ours can be found in Beltran and Krass (2002). However, our work di ers substantially from the above in that we model the decreasing nature of the rental demand, which is natural in the video rental industry. Our model accomodates any arbitrary rental duration and the dynamics of the return pattern, which is not the case in Beltran and Krass (2002). We also consider the case of endogeneous pricing and backordering of demand which has not been considered by them.

1 The base model: Exogeneous pricing with no backordering

Consider a retailer that purchases a product from a distributor at c0 per unit, rents the product to customers by charging a rental price per unit for a pre-speci ed rental duration of r periods, where r

is a positive integer. We assume that the rental demand Dt in period t is known with certainty. We only

assume that demand is decreasing in time; i.e., Dt Dt+1; 8t = 1; 2; 3; : : : ; T 1. In any period t, Dt

units are checked out from the retailer that are due back by the beginning of period t + r + 1. Out of

these Dt units that are rented in period t, let ki be the return rate so that kiDt corresponds to the total

number of units returned in period t + i. We assume that these return parameters ki are known to the

retailer

and

Pr+1

i=1

ki

=

1.

In

each

period,

returns

from

the

previous

periods

are

followed

by

realization

of

the rental demand followed by the disposal of units in the secondary market, if any.

We assume that the retailer can acquire the stock in period zero (I0) only; i.e., no replenishments are allowed in subsequent periods. We do not allow for backordering of unmet demand; i.e., demand in every period must be met. Hence It 0 8t = 1; 2; 3; : : : ; T . We also assume IT = 0 as the boundary condition, where T is the time horizon under consideration.

Let ht denotes the inventory holding cost per unit per period. Since the demand is decreasing over time and since the units are acquired only once in period zero, the retailer would want to dispose inventory in the later periods in the secondary market albeit at a discount. Let Yt 0 represents the number of units sold in period t at a price pt. We assume that the retailer incurs a xed cost Ft per period to undertake this selling activity that could include the marketing and promotional expenses. In the base model the retailer is allowed to dispose inventory in any period starting from t = 1. We assume that the disposal price is decreasing in time and that unit cost of procurement is greater than or equal to all of these prices; i.e., c0 p1 p2 : : : pT .

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1.1 Problem formulation and optimal solution structure

We write the payo to the retailer in terms of the total costs incurred over T periods. The decision

problem of the retailer can be written as:

subject to:

(

X T

= minI0;It;Yt (I0; It; Yt) = c0I0 + htIt

t=1

It; Yt; I0 0

X T

X T

ptYt + Ft (Yt)

t=1

t=1

X T

)

Dt;

t=1

It = I8t 1 Yt D^ t ; 8t 1

(1)

where

(Yt)

=

< 1; : 0;

Yt > 0; otherwise.

D^t de ned by:

8

D^ t

=

< :

Dt Dt

Pt 1 Pri=+11

i=1

kiDt kiDt

i i

t r+1 t r+2

can be intepreted as e ective demand net of all returns originating from demand in earlier periods. Since

all

demand

must

be

met

in

each

period,

PT

t=1

Dt is a constant.

This is mixed integer linear program (MILP) and can be solved using usual methods available to solve MILPs. Another option is to use dynamic programming techniques. While dynamic programming usually o ers more insights into the nature of the optimal policies, it is computationally more intensive since it involves search over a large state space. Hence, we rst employ the use of `network with directed

ows' (Denardo, 1982) to model the operations of the retail store where nodes represent time periods and ows along the arcs represent inventories, purchase and disposal quantities.

Proposition 1 There exists an optimal solution that is loopless. Proof: All proofs are provided in Deo and Tang (2005)

Proposition 1 enables us to reduce the state space for the subsequent dynamic programming formulation because we can search for the optimal solution by focusing on the set of feasible ows that satisfy the loopless property. Next, we de ne period as = argmaxft : D^t > 0g. Existence of is guaranteed since D^t < 0 8t r + 2. Given these properties, the next two lemmas characterize the properties of the optimal procurement and disposal policies, which would be used in the subsequent dynamic programming algorithm.

Lemma 2 (i) There exists a loopless solution only if s where s = argmin ft : It = 0g.

(ii) It is optimal to not sell before and during period s; i.e., Yt = 0 for 1 t s.

(iii)

The

optimal

initial

purchase

quantity

is

given

by

I0

=

Ps

t=1

D^ t.

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Lemma 3 Consider the case where Ii = 0, Ij = 0 and It > 0, s i, i < t < j. We have:

(i) D^ i+1 < 0.

(ii) If D^j < 0, then Yj =

Pj

t=i+1

D^ t

and

Yt

= 0;

i+1

t

j

1.

(iii) If D^j > 0, then there is atmost one k, such that Yk > 0, where i + 1 k j.

1.2 Dynamic programming formulation

Having obtained the necessary structural properties of the optimal solution, we now propose a dynamic programming algorithm to solve the retailer's problem optimally. Let: f (1; s) minimum cost of opera-

tion from period 1 through s, V (s) minimum cost of operation from period s to period T and c(i; j)

minimum cost of operation from period i + 1 through period j, where Ii = Ij = 0 . Then results in Lemma 2 imply that

= mins=1;2;:::; ff (1; s) + V (s)g

Lemmas 2 and 3 enable us to e ectively decompose the original problem in to two subproblems. Results

(ii)

and

(iii)

in

Lemma

2

suggests

that

f (1; s)

=

c0

Ps

t=1

D^ t

+

Ps 1

t=1

htIt,

where

It

=

Ps

i=t+1

D^ i.

Thus

for a given value of s; f (1; s) is very easy to calculate. In the second subproblem we use Lemma 3 to

formulate a dynamic program to calculate the function V (s). Note that Is = 0 in the optimal solution and IT = 0. Hence there exists at least one case in which Ii = 0 and Ij = 0. Then for any given pair (i; j) that has Ii = 0 and Ij = 0, we can formulate the second subproblem as the following dynamic program: V (i) = minj>ifc(i; j) + V (j)g. For a given i, this is exactly the shortest path problem that can be solved in polynomial time if we know the value of c(i; j). Next, we set out to nd the cost c(i; j).

Again the result in Lemma (3) can be used for this purpose.

8 >>>>>< c(i; j) = >>>>>:

+mpjiPP nktjjt==fk1ip+hk1tPDP^jtt=jl=+i+i+P11D^jtD=^til1++g;1Phtjt(=i1+P1 htl=t(i+1PD^tll=);i+1 D^ l) 1;

D^ i+1 < 0; D^ j < 0

D^ i+1 < 0; D^ j > 0 otherwise

Note that the expression of is exactly similar to that of V (i) if we expand the de nition of c(i; j) to

include f (1; s) as follows c(i; j) = f (1; s); i = 0; j = s. Then the original problem can be reformulated

as:

= V (0) Calculation of V (i) is a shortest path problem and takes O(T 2) steps if the values of c(i; j) are known. However, in this case, calculation of c(i; j) takes O(T ) steps since it involves a search over T periods. Thus it is easy to see that the complexity of the problem is O(T 3).

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2 Endogeneous pricing with no backordering

In this section we endogenize the price at which units can be disposed in the secondary market. We

assume that the quantity that can be disposed in any period is a linear function of the disposal price

Yt = at btpt. To simplify the analysis we assume that there is no xed cost per period associated with

disposals. Then the retailer's decision problem is modi ed as follows (where we have left out the rental

revenue since it is constant and not a part of the decision problem):

(

X T

1 = minI0;Yt (c0 + ht)I0

t=1

X T ( Xt ht( Yk)

t=1

k=1

))

at bt

Yt

+

1 bt

Yt2

X T (ht

Xt

D^ k)

t=1 k=1

(2)

subject to:

I0; Yt 0

Xt Yk

I0 Xt D^ k;

8t 1

(3)

k=1

k=1

where we have substituted for It = I0

Pt

k=1

Yk

Pt

k=1

D^ k

8t

1 and Yt = at

btpt.

We solve this problem by solving an inner and an outer optimization problem. In particular, we rst x I0 and solve the inner minimization problem to obtain the optimal Yt . For a xed I0, this inner optimization problem is a convex (quadratic) optimization problem with linear inequalities and hence can be solved to optimality using conventional methods of convex programming. We derive the dual of this problem which is a convex (quadratic) program with tree constraints and can be solved in polynomial time (Tang, 1990). The reader is referred to Deo and Tang (2005) for further analysis and managerial insights into this problem.

3 Extensions to the base model

In the extension of the base model, we allow multiple purchase opportunities. In this case, we nd that there exists an optimal solution that is loopless. However, in contrast to the conventional model with multiple purchases and no returns, c.f., Denardo (1982), we nd that the periods of zero inventory and periods of purchase need not coincide in the optimal policy. This is because of the possibility of negative e ective demand; i.e., returns from earlier periods are greater than the rentals during that period. Hence even though the inventory at the end of a period might be zero, the e ective demand in the next period might be negative and hence it is pro table to postpone the purchase to a later period and purchase at a lower cost.

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4 Concluding Remarks

In this paper, we have presented various models to obtain optimal procurement and disposal policies of a rental retailer with speci c focus on the video rental business. We considered cases with exogeneous and endogeneous prices and with and without backordering and the case with multiple procurement opportunities. The rental demand in practice is not deterministic, as assumed in the model. However, a good forecast can be obtained using box-o ce collections and DVD sales. Future research could explore extension of the model to uncertain demand. See Tang and Deo (2004) for a possible model. Other possible extensions could include variable rental price due to movement of the title from \new releases" to \catalog" section and cannibalization of the rental revenue as a result of the disposal policy of the rental retailer.

References

Beltran, J.L., and D. Krass. 2002. Dynamic Lot Sizing with Returning Items and Disposals. IIE Transactions 34 (5) 437-448.

Cachon, G., and M. Lariviere. 2005. Supply Chain Coordination with Revenue Sharing Contracts: Strengths and Limitations. Management Science 51 (1) 30-44.

Dana J., and K. Spier. 2001. Revenue Sharing and Vertical Control in the Video Rental Industry. Journal of Industrial Economics 59 (3) 223-245.

Dana, J. 2001. Competition in Price and Availability when Availability is Unobservable. Rand Journal of Economics 32 (4) 479-513.

Denardo, E. 1982. Dynamic Programming: Models and Applications, Prentice Hall, Inc., Englewood Cli s, New Jersey.

Deo, S. and C.S. Tang. 2005. Optimal Procurement, Disposal and Pricing Policies for Managing Rental Goods. Working paper UCLA Anderson School of Management.

Mortimer, J. 2004. Vertical Contracts in the Video Rental Industry. Working paper Department of Economics, Harvard University.

Tang, C.S. 1990. Reducing separable convex-programs with tree constraints. Management Science 36 (11) 1407-1412.

Tang, C.S., and S. Deo. 2004. Issues Arising from Managing a Rental Store: Rental Duration, Inventory, Rental Price and Competition. Working paper UCLA Anderson School of Management.

Wagner, H.M., and T.M. Whitin. 1958. Dynamic Version of the Economic Lot Size Model. Management Science Vol. 5 (1) 89-96.

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