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Lisa Gimpl-Heersink and Christian Rudloff and Moritz Fleischmann andAlfred Taudes

Integrating Pricing and Inventory Control: Is it Worth the Effort?

Article (Published)(Refereed)

Original Citation:Gimpl-Heersink, Lisa and Rudloff, Christian and Fleischmann, Moritz and Taudes, Alfred (2008)Integrating Pricing and Inventory Control: Is it Worth the Effort? Business Research, 1 (1). pp.106-123. ISSN 2198-3402

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BuR -- Business ResearchOfficial Open Access Journal of VHBVerband der Hochschullehrer für Betriebswirtschaft e.V.Volume 1 | Issue 1 | May 2008 | 106--123

Integrating Pricing and Inventory Control:Is it Worth the Effort?Lisa Gimpl-Heersink, Institute for Production Management, Vienna University of Economics and Business Administration,Email: [email protected]

Christian Rudloff, Institute for Production Management, Vienna University of Economics and Business Administration,Email: [email protected]

Moritz Fleischmann, Rotterdam School of Management, Erasmus University, Email: [email protected]

Alfred Taudes, Institute for Production Management, Vienna University of Economics and Business Administration,Email: [email protected]*

AbstractIn this paper we first show that the gains achievable by integrating pricing and inventory controlare usually small for classical demand functions. We then introduce reference price models anddemonstrate that for this class of demand functions the benefits of integration with inventory controlare substantially increased due to the price dynamics. We also provide some analytical results forthis more complex model. We thus conclude that integrated pricing/inventory models could repeat thesuccess of revenue management in practice if reference price effects are included in the demand modeland the properties of this new model are better understood.

Keywords: Inventory Management, Pricing, Reference Price, Stochastic Demand Model

1 Introduction

1.1 Problem Statement

In the basic price-optimization problem, the onlyinternal factor affecting the price of a monopo-list is incremental cost, i.e. profit-optimal pricep* is given by p* = cη(p*)/(η(p*) − 1), whereη(p*) is the price elasticity and c ≥ 0, incre-mental cost (see, e.g., Phillips (2005), Chapter 1).In the case of linear demand D(p) = β0 + β1pwith β0 > 0,β1 < 0, optimizing (p − c)D(p) yieldsp* = (β1c−β0)/2β1 as the optimal price. This resultis valid if enough inventory (capacity) is in placeto satisfy the resulting demand D(p*) or if demandis deterministic and D(p*) is produced/procuredafter the pricing decision before the selling pro-cess starts. Inventory/capacity enters the pricingproblem if the available inventory is smaller thanD(p*): in this case the optimal price is the run outprice pr,D(pr) = x, where x is the available inven-tory (capacity). Profit can be improved if differentprices are charged to different market segments.This is done by either adjusting price over time

* This research was supported by the Vienna Science andTechnology Fund (WWTF) under grant ‘‘Integrated Demandand Supply Chain Management’’ (Call ‘‘Mathematics and ...’’).

(so-called price-based revenue management) orby dynamically closing booking classes (so-calledquantity-based revenue management) dependingon the inventory/capacity remaining for the rest ofthe selling period (see e.g. Talluri and van Ryzin(2004)). This kind of integrated pricing and inven-tory (capacity) management has a profound impacton profitability and revenue management is ac-cepted industry practice in the service sector (yieldmanagement for airlines, hotel, logistics serviceproviders etc.) and retailing (markdown manage-ment of high-tech/fashion goods or sell-off items).In this paper we investigate the situation whereinventory is adjustable, i.e. the inventory decisionis made after or jointly with the pricing decisionbefore demand materializes. Demand is uncertain,i.e. D(p, ε) = β0 + β1p + ε, where ε follows a distri-bution function F(·) with mean 0 and variance σ2.This is typically the case in manufacturing or inretailing of goods with long life cycles and shortprocurement/production lead times1.In many firms, pricing and inventory control isdone sequentially in such situations. At first, mar-

1 We will thus only use the term inventory in the following,although the findings also hold for suitable settings in serviceindustries, where the analogous concept is capacity.

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keting/sales autonomously determines a price thatmaximizes expected profit based on incrementalcosts c only. This is then passed on to logistics(procurement or production) together with infor-mation on the demand distribution. Logistics thenadds a safety stock to expected demand E[D(p*, ε)]to account for risk depending on shortage, holdingand backlogging cost and salvage value. However,while computationally simple and well adapted tothe usual division of labor in firms, in general theresulting prices and inventory policies are not opti-mal. As it was already shown in Whitin (1955), theoptimal price resulting from a joint optimizationof a price-setting Newsvendor Problem with lostsales differs from p*. It seems natural to acceleratedemand through a price discount in the case ofoverstocking in a multi-period setting. However,these gains have the prize of added complexity andintra-organizational coordination cost and practicewill only adopt more elaborate models and busi-ness processes if the expected benefits outweighthis overhead. Classical works on integrated pric-ing and inventory control indicate profit increasesthrough integration of only up to 2 % for typicalcases, which might be an explanation of why --contrary to revenue management -- such modelsare not widely implemented in IT-based planningsystems so far.

1.2 Existing Literature and Contributionof the Paper

Yano and Gilbert (2004) and Elmaghraby andKeskinocak (2003) survey the literature on com-bined pricing/inventory control. Regarding the de-mand model, two streams of literature can bedistinguished: the works in the operations man-agement/logistics area assume classical demandmodels, where demand is influenced by the cur-rent price only. Here, the focus is on details of thelogistic part. Inferences about the gains of integra-tion are either done analytically (see, e.g., Petruzziand Dada (1999)) or via numerical simulation (see,e.g., Federgruen and Heching (1999)). As statedabove, these results show limited benefits.Reference price models have been popular in mar-keting (see e.g. Mazumdar, Raj and Sinha (2005)for a survey). Here also the pricing history influ-ences demand, as it is the basis of the expectationof future prices and acts as a benchmark to judgecurrent prices: customers respond favorably if the

current price is less than the reference price andunfavorably if the current price is higher than thereference price. To the best of our knowledge, theintegration of reference price models with inven-tory control models has not been studied so far.We will probe into the issue of whether using a ref-erence price model to describe demand increasesthe benefits of integration with a logistic model.Here, we study whether the base-stock list-pricepolicy still holds in such a setting and infer thesize of potential benefits via numerical simulation.In the survey part at the beginning of the paperwe also complement existing literature by studyingsingle-period joint pricing/inventory control withbacklogging and by extending the steady-state ref-erence price proof from Popescu and Wu (2007)to the case of positive ordering cost.

1.3 Outline of the Paper

In Section 2, we will briefly describe the sequen-tial method in a single- and multi-period settingas the basis for a review of the integrated pricingand inventory decision models developed in liter-ature and study the benefits of an integration inSection 3. In Section 4 we then extend the clas-sical demand model by including reference priceeffects. In Section 5 we summarize our findingsand identify areas for further research.Generally, our focus is on linear demand modelsand monopolistic pricing. While ensuring math-ematical tractability these assumptions are notunrealistic: price optimization by a firm is onlypossible in imperfect markets and in the case ofmonopolistic competition a firm faces a range ofprices where competitors do not react, i.e. thelinear demand function is a local approximationconditional on competitors’ prices which remainunchanged if the price stays within this permis-sible range (see Phillips (2005), Chapter 1). Forthe logistic part we only look at the determinationof the inventory/production level. Consistent withthe inventory literature we will therefore denote cas ordering cost. We also focus on the backloggingcase,2 where unsatisfied demand is fully satisfiedin later periods and backlogging costs are chargedand assume no fixed ordering costs.

2 A line of related research concentrates on partial backlog-ging either for non-perishable goods (see e.g. Abad (2001)) orperishable goods (see e.g. Dye, Hsieh and Ouyang (2007)).

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2 Sequential Models

2.1 Newsvendor Models

In the case of a perishable product that becomesobsolete at the end of a single period and whereunsatisfied demand is lost, in the sequential modellogistics solves the well-known classical Newsven-dor Problem (see, e.g., Petruzzi and Dada (1999)).In the first step marketing/sales determines theprofit-optimal price as

(1) p* = −β0 − β1c2β1

due to the fact that E[ε] = 0. Then, p* and infor-mation on demand risk is passed on to logistics,which determines the optimal inventory level y* as

(2) y* = E[D(p*, ε)] + F−1

(p* + s − cp* + s − v

),

where s ≥ 0, denotes the shortage cost and v thesalvage value (or salvage cost ifv < 0, respectively).Here Cu = (p* + s − c) represents the opportunitycost of underestimating demand and Co = (c − v)the cost of overestimating demand. The above ratioCu/(Cu +Co) in equation (2) is known as the criticalfractile. Intuitively, it corresponds to the safetyfactor at which the expected profit lost from beingone unit short is equal to that from being one unitover.Logistics employs a different model if demand inexcess of the amount stocked is backlogged. In thiscase, customers will return after the end of theperiod where there is one more chance to placean order for the outstanding items, which are theninstantaneously delivered to the customers3. But,at the same time, backlogging costs b ≥ 0, arecharged as penalty costs for the inability to meetconsumer needs when they occur. In cases of over-production with resultant excess of inventories atthe end of the period, holding costs h ≥ 0 occur.These could be interpreted as carrying charges un-til the remaining items can be sold e.g. to a discountstore for some salvage value v. Holding and back-logging costs are charged for the period when theyoccur, whereas any financial flow after the end ofthe period (reordering/salvaging opportunity) isdiscounted by a discount factor 0 < γ ≤ 1. To in-sure that it is not optimal to not order anything at

3 Problems like these can be found as newsvendor models withan emergency supply option (see e.g. Montgomery, Ajit andKeswani (1973) and Khouja (1996)).

all and merely accumulate backlog penalty costs,also b > (1 − γ)c is assumed. Maximizing expectedprofits yields the optimal order quantity

(3) y* = E[D(p*, ε)] + F−1

(b − (1 − γ)c

h + b − γ(v − c)

).

To see this we differentiate the expected one-periodprofit

E[P(y,p*)] = p*E[D(p*, ε)] − cy(4)

− (h − γv)

y−E[D(p*,ε)]∫−∞

(y − E[D(p*, ε)] − u)f (u)du

− (b + γc)

∞∫y−E[D(p*,ε)]

(E[D(p*, ε)] + u − y)f (u)du

with respect to the inventory y. This leads to

∂E[P(y,p*)]∂y

(5)

= −c − (h − γv)F(y − E[D(p*, ε)])

− (b + γc)F(y − E[D(p*, ε)]) + (b − γc) .

Setting this equal to zero and solving for y givesthe desired result.The critical fractile (b − (1 − γ)c)/(h + b − γ(v − c))has a similar interpretation as in equation 2: itcorresponds to the order quantity at which theexpected profit lost from being one unit shortis equal to that from being one unit over. HereCu = (b − (1 − γ)c) denotes the opportunity costof underestimating demand and Co = (h + c −γv) the cost of overestimating demand. The aboveratio is again given by Cu/(Cu + Co). Note that inthe backlogging case p* does not appear in thecritical fractile. This is because the items are soldin any case. Today, the situation characterized byformula (3) is more prevalent in practice, as due tocompetition, firms are willing to incur substantialbacklogging cost rather than lose customers in thefuture due to unsatisfied demand.

2.2 Multi-Period Models

We now turn to the dynamic version of the backlog-ging inventory model which was first introducedand solved by Arrow, Harris and Marschak (1951).Here the system described will be operated over Tperiods. What makes the problem more compli-cated than solving T copies of the single-periodproblem is that any leftover stock at the end of

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Figure 1: Inventory Path

one period is retained and can be offered for salethe following period (see Figure 1 for a sampleinventory path). In this regard, each unit of posi-tive leftover stock at the end of each period incursholding costs h. Demand for the product in excessof the amount stocked will be backlogged, whichmeans that these customers will return the nextperiod for the product, in addition to the usualrandom number of customers. A per-unit backlog-ging cost b is charged as a penalty cost for each unitthat is backlogged in each period. The newly aris-ing demands in different periods are assumed to bestatistically independent. As in the above sectiona proportional per-unit ordering cost of c is in-curred, and orders placed are essentially receivedimmediately (received in time to meet demandthat arises in that period). All costs are expressedin beginning-of-period cash units, cash flows oc-curring in subsequent time periods are discountedby a one-period discount factor γ ∈ (0,1]. Afterthe last period, the remaining inventory is salvagedor backlogged demand is satisfied.A convenient way to represent the state of a systemin this model is the level x of inventory before or-dering. Positive x indicates leftover stock at the endof the period, negative x unmet demand of the pre-vious period (see Figure 1). The objective is to findan optimal order-up-to level yt for each period tin order to maximize total expected discountedprofits:

T∑t=1

γt−1maxyt

(p*t E[D(p*

t , εt)] − c(yt − xt)(6)

− G(yt,p*t )

)+ γTL(yT ,p*

T ) ,

with D(p*t , εt) = β0 + β1p*

t + εt, where β0 also in-cludes the effect of demand drivers other than price(e.g. trend, seasonality, promotions, etc.),

Figure 2: Optimal inventory level after ordering

G(yt,p*t ) = E[h · max((yt − D(p*

t , εt)),0)

+ b · max((D(pt*, εt) − yt),0)]

are the expected inventory holding/backloggingcosts andL(yT ,p*

T ) = E[v·max((yT−D(p*T , εT )),0)+

c · max((D(p*T , εT ) − yT ),0)] the expected salvage

value (costs)/ backlogging costs in the last pe-riod T . Note that we are only using the time-invariant demand and inventory costs. Many re-sults are also possible in the time variant case butare omitted for clarity of the formulas.This problem is mostly stated in literature in termsof dynamic programming:

Vt(xt) = maxy

{p*t E[D(p*

t , εt)] − c(y − xt)(7)

− G(y,p*t ) + γE[Vt+1(y − D(p*

t , εt))]}

,

with VT+1 = L(yT ,p*T ).

Bellman, Glicksberg and Gross (1955) were the firstto show that the optimal cost function is convexin inventory before ordering xt, which leads tothe result that a stock level independent of theinventory level before ordering is optimal. Thislevel is often referred to as the base-stock level. Ifthe initial inventory level xt is below the base-stocklevel, an order is placed to raise the inventorylevel yt to the base-stock level St, otherwise, noorder is placed (see Figure 2).4

Figure 3 shows that for the finite horizon case withno salvage value, the optimal base-stock level is

4 E.g. Zheng (1991) extends this result to a case of additionalfixed ordering costs. Using the property of k-convexity theyshow that an (s, S)-policy is optimal, where the order-up-tolevel is again given by a constant, but an order is only placedif the inventory level before ordering is below another constantlevel.

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Figure 3: Base-stock path over time

decreasing over time. That is because towards theend of the planning horizon, since time remainingis getting shorter, the risk of not selling the in-ventory on stock is increasing, against which coststhe decision maker hedges by a diminishing base-stock level. Of course no risk is observed in thecase where the per unit salvage value equals theper unit ordering costs (v = c). In this case, if someinventory on stock is not sold by the end of thehorizon, it can be salvaged by the same amountof money as it was ordered. Here, a myopic policywhich looks only at the single-period backloggingproblem described in Subsection 2.1 is optimal inevery period, regardless of the time horizon T . Inother words a steady-state solution for the infinitehorizon case can be simulated by setting v = c. Thesteady-state base-stock is given by

(8) y* = E[D(p*, ε)] + F−1

(b − (1 − γ)c

h + b

).

This can be shown analytically by rearrangingterms and extending equation (6) to the infinitehorizon case in the following way. We replacext+1 = yt − D(pt) for all t ≥ 1 and then rearrangethe sum in such a way that we are left only withterms indexed by t + 1 in the t-th summand:

V (x1) =∞∑t=0

γt maxyt+1

[p*E[D(p*, εt+1)](9)

− c((yt+1 + xt+1) − G(yt+1,p*)]

= cx1 +∞∑t=0

γt maxyt+1

[p*E[D(p*, εt+1)]

− c((1 − γ)yt+1 + γD(p*)) − G(yt+1,p*)]

.

As we assume that we are in a steady state we re-place the time-dependent yt by in time-invariant y.Since the profit function V (x1) is a concave func-tion in y, the optimal inventory level after or-dering y* can now be obtained by differentiat-ing V (x1) with respect to y and setting the resultequal to zero. Using Leibniz integration rule we candifferentiate the expected inventory costs to get∂∂yG(y,p*) = −b + (h + b)F(y − ED(p*)) where F(·)is the cumulative distribution function of the de-mand’s random perturbation ε. Hence, to find thesteady-state solution y*, we solve

∂∂y

V (x1) =1

1 − γ

(− (1 − γ)c + b(10)

− (h + b)F(y − ED(p*)))

= 0 ,

which results in equation (8).Table 1 displays the optimal steady-state base-stock level S*, which is the optimal order-up-tolevel, if the inventory before ordering is smallerthen S*. We investigate the effect on the optimalsolution for various parameter settings. The baseparameters are: β0 = 100, β1 = −20, h = 0.005,b = 0.4, σ = 40, γ = 1, D(p, ε) ~ normal. Accord-ing to Phillips (2005) we set the price to p* = 2.75.Since F−1(·) is increasing in both discount fac-tor γ and backlogging costs b, the steady statebase-stock level S* is also increasing in these pa-rameters. An intuitive explanation for the changesin backlogging costs could be that the seller wantsto hedge against higher backlogging costs by keep-ing higher inventory levels. Furthermore, a higheruncertainty in demand also results in higher safetystock levels and hence in higher base-stock lev-els. This is illustrated in Table 1 by varying oneof the parameters in each row. Numerical resultsalso show that for more heavy tailed distributeddemands (like the beta or the log-normal distribu-tion) base-stock levels are higher to prepare for thehigher risk of large demands.

Table 1: Steady-state base-stock level S*

b 0.05 0.09 0.16 0.4

S*b 40 64 81 103

σ 10 20 40 60

S*b 59 74 103 132

γ 0.75 0.85 0.95 1

S*b 64 79 103 135

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3 Classical Joint Pricing andInventory Control Models

3.1 The Price Setting NewsvendorProblem

We now look at the Newsvendor Problem whenprice and inventory level are jointly optimized. Wewill start with the lost sales case, which is studiedin Whitin (1955), Thomas (1974) and surveyedin Petruzzi and Dada (1999). If we define the safetystock z = y − E[D(p, ε)], the goal function is

E[P(z,p)] =(11) ∫ z

−∞(p(E[D(p, ε)] + u) + v(z − u))f (u)du

−∫ ∞

z(p(E[D(p, ε)] + z) − s(u − z))f (u)du

− c(E[D(p, ε)] + z)

As Petruzzi and Dada (1999) show, E[P(z,p)] isconcave in z for a given p and concave in p fora given z. One can thus reduce the optimizationof (11) to an optimization problem in a single vari-able, first solving for the optimal value of z asa function of p and then substituting the resultinto E[P(z,p)], which yields the fractile rule fromequation (2), or by first optimizing p for a given zand then searching over the resulting optimal tra-jectory to maximize E[P(z,p)]. In this case theoptimal price for the integrated problem is givenas (see Petruzzi and Dada (1999))

(12) p*1 = −

β0 − β1c2β1

+Θ(z)2β1

,

where Θ(z) =∫ ∞z (u − z)f (u)du denotes the ex-

pected lost sales when safety stock z is chosen.Thus, since β1 < 0, the sequential method yieldsa price that is higher than the optimal price forthe integrated problem and a gain can be achievedby more closely coordinating marketing and logis-tics. In the integrated setting the price is used toreduce the coefficient of variation of demand, andthe difference between the optimal price set bymarketing in isolation is decreasing with increasedprice sensitivity (slope of the demand function β1)and demand uncertainty. However, as Petruzziand Dada (1999) show, this effect is reversed ifrandomness is modelled in a multiplicative way asD(p) = (β0 + β1p) · ε.Let us now look at the backlogging case. If we fixz = y − E[D(p, ε)], the price does not influence

holding/backlogging costs anymore. This differsfrom the lost sales case, where we lose sales ifdemand exceeds stock and hence revenue getsreduced by p · (ε − z). Differentiating the expectedprofit (4) with respect to price yields

(13)∂E[P(y,p)]

∂p= E[D(p, ε)]+(p−c)E[Dp(p, ε)],

where E[Dp(p, ε)] = ∂E[D(p,ε)]∂p . Setting (13) equal to

zero and solving for p leads to p* as given in (1) be-ing optimal, i.e. in the backlogging case, we can setthe price independently of the inventory decision.Thus, while the sequential approach is not optimalin the lost-sales case, no gain is achieved by jointoptimization in the backlogging case.

3.2 Multi Period Joint Pricing andInventory Control

The multiple period joint pricing and inventorycontrol problem has been studied e.g. by Feder-gruen and Heching (1999), Chen and Simchi-Levi(2004a) and Chen and Simchi-Levi (2004b). Herein each period the level of inventory before order-ing x is observed and on this basis both y and pare set, with the intuition that holding costs can bereduced by accelerating demand via reducing p.LetVt(xt) be the maximum expected profit from pe-riod t onwards (profit-to-go function), with initialinventory xt. Then the recursive Bellman equationhas the following form:

(14) Vt(xt) = maxyt≥xt ,pt

{Jt(xt,yt,pt)}

Jt(xt,yt,pt) = E[Pt(xt,yt,pt, εt)](15)

+ γE[Vt+1(yt−D(pt, εt))

]with Pt(xt,yt,pt, εt) = pt(β0+β1pt +εt)−c(yt −xt)−G(yt,pt) and the boundary condition V *

T (xT ) =0 for all xT . Following Federgruen and Heching(1999), we reformulate (14) and (15) such thatVt(xt) is reduced by cxt with G(yt,pt) being definedas in (6):

(16) Vt(xt) = maxy≥x, p

Jt(yt,pt)

Jt(yt,pt) = (pt − γc)E[D(pt)

](17)

− c(1 − γ)yt − G(yt,pt)

+ γE[V *t+1(yt − D(pt, εt)

]and VT (xT ) = cxT for all xT .

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For this model, Federgruen and Heching (1999)use joint concavity to show optimality of a base-stock policy and submodularity5 in y and p to showoptimality of a list-price policy. That means thatin time period t, if the current inventory level xt isbelow the base-stock, inventory is increased to ytand the price p*

t is charged. Otherwise, no orderis placed and a discount that increases with xt isgiven. Here, p*

t is called list price, as it resemblesthe price level without discounting.Similar to the steady state of model (6) one can finda steady state for the joint pricing and inventorycontrol model (14) and (15). For the linear demandcase the steady states are given as follows: thesteady-state price is

(18) p* = −β0 − β1c2β1

while the steady-state base-stock is

(19) y*(p*) = E[D(p*, ε)] + F−1

(b − (1 − γ)c

h + b

).

Note that the optimal steady-state price p* equalsthe optimal price from Section 1.1. The proof worksexactly like the proof in Section 2.2. Again werearrange the infinite sum (compare formula (9))and replace the time variant variables yt and pt bytheir time-invariant counterparts y and p. In orderto find the steady-state base-stock y*(p*) (this timedepending on the optimal steady state price p* ofthe joint pricing and inventory control model) wefollow the steps from Section 2.2 to obtain theoptimal steady-state base-stock (19). To find thesteady-state price, we use y*(p) as an input for y inthe infinite sum representation of formula (15) anddifferentiate with respect to p. To find p* we haveto make y* dependent on p instead of p* and thenoptimize with respect to p in order to obtain p*.It is easy to see that in this case, G(y*(p),p) isa constant and does not depend on p anymore. AsV (x1) is concave in p, we need to solve

∂∂p

V (x1) = E[D(p, ε)] +∂E[D(p, ε)]

∂p(p − c)(20)

= 0 .

In the case of the linear demand function thisyields equation (18). Note that the optimal steady-

5 For all y1 > y2, J(y1, p) − J(y2, p) is non-increasing in p.

Figure 4: Price trajectory for time period 5, 10 and 15

Figure 5: Base-stock and list-price evolution over time

state price is the same as the optimal myopic pricefrom Phillips (2005).Figures 4 and 5 depict numerical results for (16)and (17) for a parameter setting similar to Fe-dergruen and Heching (1999) with β0 = 100,β1 = −20, c = 0.5, b = 0.4, h = 0.005, v = 0,T = 15 and a coefficient of variation of 0.44.Figure 4 shows the price as a function of the inven-tory before ordering for different periods. Figure 5shows the base-stock, price and expected demandfor the last periods of the planning horizon. Onecan see that while the base price stays constant overtime, the tendency to give price discounts to lowerinventory increases over time. Figure 6 depicts thegains of applying equations (16) and (17) versusthe sequential procedure (1). We get the largestbenefits of joint optimization towards the end of,or for a short planning horizon. In contrast to thecomparisons in Federgruen and Heching (1999),

112


Figure 6: Profit increase over time

Figure 7: Profit increase in inventory level x0

who base all numerical results on a coefficient ofvariation, we always use a constant standard devi-ation (not depending on price). This erodes a lot ofthe benefit when using a dynamic pricing model.Figure 7 shows relatively low benefits for low stockbefore ordering, which can be much higher forsubstantially larger inventory levels before order-ing. The closer we get to the end of the planninghorizon, the earlier this effect can be observed.This is intuitive as the seller tries to reduce the riskof being left with unsold stock at the end of theplanning horizon.

4 Joint Pricing and InventoryControl with Reference PriceEffects

4.1 Reference Price Models

Empirical studies like e.g. Winer (1986), Greenleaf(1995), Kopalle, Rao and Assuncao (1996), Bri-

esch, Krishnamurthi, Mazumdar, and Raj (1997),Fibich, Gavious and Lowengart (2003), Mazum-dar, Raj and Sinha (2005) and Natter, Reutterer,Mild, and Taudes (2006) show that for goods whichare repeatedly bought from a monopolist, demandnot only depends on the current price but is alsosensitive to the firm’s pricing history and thusaccounts for intertemporal demand correlations.Since consumers have a memory, the carrier ofprice is not only based on its absolute level, butrather on its deviation from some reference levelresulting from the pricing history. As customersrevisit the firm, they develop price expectations,which become a benchmark against which currentprices are compared. A formulation that capturesthis effect is the so-called reference price which isa standard price against which consumers evaluatethe actual prices of products they are considering.If the price is below the reference price, the ob-served price is lower than anticipated, resultingin a perceived gain. This would make a purchasemore attractive and raise demand. Similarly, theopposite situation would result in a perceived loss,reducing the probability of a purchase (people areless likely to buy products after prices have goneup). To include reference price effects, our lineardemand model thus becomes (see e.g. Greenleaf(1995) or Kopalle, Rao and Assuncao (1996)):

D(pt, rt, εt) = β0 + β1 · pt(21)

+ β2 · max{pt − rt,0}

+ β3 · min{pt − rt,0} + εt ,

with β0 ≥ 0, β1,β2,β3 ≤ 0 and εt being iid. accord-ing to an arbitrary probability density function f (·)with mean E[εt] = 0. Price and reference price rtare restricted to an arbitrary finite interval [p,p]and [r, r] such that E[D(pt, rt, εt)] ≥ 0.If equation (21) is symmetric with respect to theeffect of gains and losses (β2 = β3), buyers areloss-neutral and the demand function is smooth.For loss-averse consumers the demand function issteeper for losses than for gains (β2 < β3). In otherwords, a loss decreases expected demand morethan an equivalently sized gain would increase de-mand (see Figure 8). This behavior is predicted byProspect Theory (see e.g. Winer (1986)). If β2 > β3,consumers are loss-seeking. As Slonim and Gar-barino (2002) show, β2 > β3 can also arise on theaggregate level when in fact the consumers behaveaccording to Prospect Theory but stockpile when

113


Figure 8: Prospect theory

Figure 9: Formation of reference price

prices are low. We will focus on the loss-neutraland loss-averse case, which yield closed-formedsteady-state solutions, while the optimal pricingpolicy in the loss-seeking case cycles (see Popescuand Wu (2007) and Figure 14).A commonly used simple reference price updatingmechanism is

(22) rt = αrt−1 + (1 − α)pt−1 ,

where 0 ≤ α < 1 denotes the memory parameterand captures how strongly the reference price de-pends on past prices6 (see Figure 9). Lower valuesofα represent a shorter term memory; in particularif α = 0 the reference price equals the price of theprevious period.

6 For a different, detailed exposition of reference price mecha-nisms, see e.g. Moon, Russell, and Duvvuri (2006).

In such a setting, the objective is to maximize totalexpected profits:

(23) maxpt∈[p,p]

T∑t=1

γt−1(pt − c)E[D(pt, rt, εt)

].

The decision which price to charge in each periodis made in stages and cannot be viewed in isolationsince one must balance the desire for high presentprofits, obtained by charging relatively low prices,against the undesirability of low future profits, re-sulting from the formation of a low reference priceas a consequence of the earlier price discount. Thistradeoff is captured by the technique of dynamicprogramming which at each stage ranks decisionsbased on the sum of the present values and theexpected future values, assuming optimal decisionmaking for subsequent stages. As the referenceprice summarizes past information which is rele-vant for future optimization, equation (23) is oftenwritten in terms of the Bellman equation:

Vt(rt) = maxp∈[p,p]

{E

[(p − c)D(p, rt, εt)

](24)

+ γE[Vt+1(αrt + (1 − α)p)

] }.

An important consequence of this reference priceformation is that although frequent price discountsmay be beneficial in the short run, they may be dan-gerous in the long run when consumers get used tothese discounts and reference prices drop. The re-duced price becomes anticipated and loses its effec-tiveness, whereas the non-promoted price becomesunanticipated and would be perceived as a loss.Kopalle, Rao and Assuncao (1996) and Popescuand Wu (2007) show for time-constant parame-ters that if the reference level is initially high, anoptimizing firm should consistently price belowthis level, which has the effect of a skimming strat-egy. Similarly a low initial reference level leadsto the optimality of a penetration type strategy.According to Popescu and Wu (2007) the optimalprice path converges (monotonously under certainassumptions) to a constant steady state which theyonly give for the case of zero ordering costs. Ex-tending their results to the case c > 0 leads to thefollowing result for the the optimal steady-stateprice under the piecewise linear demand modelof equation (21) for loss neutral (β2 = β3) or lossaverse (β2 < β3) customer behavior:

114


(25)

p*(r1) =

⎧⎪⎪⎨⎪⎪⎩p*p = (β1c−β0)(1−αγ)+β2(1−γ)c

2β1(1−αγ)+β2(1−γ) if r1 ≤ p*p

p*s = (β1c−β0)(1−αγ)+β3(1−γ)c

2β1(1−αγ)+β3(1−γ) if r1 ≥ p*s

r1 else.

Here, p*p is the steady-state solution obtained by

a penetration type strategy (if initial price ex-pectations are low, start with a low price andmonotonously increase it until the steady stateis reached) and p*

s is the steady-state solution ob-tained by a skimming type strategy (if initial priceexpectations are high, start with a high price andmonotonously decrease it until the steady state isreached).Like in the steady-state proofs before, we againuse differentiation of the infinite case of equa-tion (23). We substitute pt = p and rt+1 = αt(r1 −p) + p for all t, where r1 is the starting referenceprice. In the linear demand case, this results inV (r1) =

∑∞t=0 γ

t[(p − c)(β0 + β1p − αtβ2(r1 − p))

].

Differentiating V (r1) with respect to p and settingequal to 0 yields

2β11 − γ

p +β2/31 − αγ

(2p − r1)(26)

+β0 − cβ11 − γ

−β2/3c1 − αγ

= 0 .

Using the assumption that the reference priceis in a steady state r we set rt = r for all t.Hence, from equation (22) it follows that r1 = p.Since V (r1) is concave, solving this equation yieldsthe steady-state price (25) for the loss neutralcase. An extension for the proof to non-lineardemand functions and the loss averse case canbe found in Popescu and Wu (2007). Note, thatif (β1c−β0)(1−αγ)+β2(1−γ)c

2β1(1−αγ)+β2(1−γ) < r1 < (β1c−β0)(1−αγ)+β3(1−γ)c2β1(1−αγ)+β3(1−γ) ,

we have p*(r1) = r1. As can be seen from equa-tion (25), the steady-state price is the optimal pricewithout reference price effects if α = 0 and γ = 0and β2/3 = 0. Popescu and Wu (2007) also showthat the steady-state price is decreasing in α and γand increasing in β2/3. That is, it is the lower thesmaller the reference price effect is. We use a basescenario of the demand model given in eq. (21)and set the parameters for loss neutral customerbehavior to β0 = 100,β1 = −20,β2 = β3 = −40and β2 = −50,β3 = −30 for loss averse customers,respectively. Unless not stated differently we as-sume α = 0.5, γ = 0.5. Table 2 shows how the

Table 2: Steady-state list price level p*

β2 −40 −20 0

p*β2

4.3 4.38 4.5

γ 0 0.33 0.66

p*γ 4.25 4.28 4.33

α 0.66 0.33 0

p*α 4.29 4.31 4.33

steady-state price p*(r1) is influenced by succes-sively varying the reference effect β2 = β3, thediscount rate γ and the memory parameter α.Letting γ = 0 equation (25) describes the one-period profit-optimizing price which we denoteby p̃*(r1). By simple algebra and the fact thatβ0 + β1c ≥ 0 (since p ≥ c) it is easy to see thatp̃*(r1) ≤ p*(r1), stating that the prices charged bya myopic firm are oblivious of their eroding effecton future demand, hence future profits. Denotingthe optimal price in absence of a reference priceby p̂* it is also easy to show that p*(r1) ≤ p̂* for anyr1 ≥ 0. This means that in the long run, strategicfirms should charge lower prices when consumersform reference effects, than when they do not.Bounds for the prices in Table 2 are p̃* = 4.25 andp̂* = 4.5.From equation (25) it also becomes clear that in theloss averse case there exists more than one steadystate, depending on the initial reference price r1.For initial price expectations lying between the twopossible steady states p*

p and p*s, a constant pricing

policy of the customer’s initial price expectation r1is optimal (see Figure 10). Sample price paths forthe loss neutral, loss averse and loss seeking case

Figure 10: Optimal price path (loss averse)

115


Figure 11: Optimal price and reference price path

Figure 12: Optimal and myopic pricing policies

are given in figures 11 to 14. Moreover, for thoseFigures we set c = 4and restrict price and referenceprice to the interval p ∈ [4.2,4.4], r ∈ [4.2,4.4]ensuring nonnegative demand for any p and r.Note that in the loss seeking case (see Figure 14)the optimal policy is cycling and not allowing for ananalytical solution (see Popescu and Wu (2007)).

4.2 Joint Pricing and Inventory Controlunder Reference Price Effects

Not much is changed if reference effects are intro-duced in the one-period backlogging case: as theoptimal inventory level is independent of the price,r is only an additional parameter so that for thelinear demand function the optimal price changesto

(27) p*1(r) =

−β0 + cβ1 + β2/3(r + c)2(β1 + β2/3)

,

Figure 13: Optimal steady-state price (loss averse)

Figure 14: Optimal price path (loss seeking case)

which is less than p* from the joint pricing andinventory control model without reference effects(see equation (18)) and increasing in the referenceprice r. The optimal order-up-to level y* is thengiven by (3) where E[D(p*, ε)] is replaced by theexpected demand depending on the reference priceas well. Not surprisingly, it can be seen that p*(r) isincreasing in r, since a higher price can be chargedwhen the customer is expecting it.In the multi period case we now have a state spaceconsisting of two variables, the initial inventory xand the reference price r. Thus, the profit-to-gofunction is specified as V *

t (xt, rt) and the recursiveBellman equation has the following form:

(28) V *t (xt, rt) = max

yt≥xt ,pt{Jt(xt,yt,pt, rt)}

Jt(xt,yt,pt, rt) = E[Pt(xt,yt,pt, rt, εt)](29)

+ γE[V *t+1(yt−D(pt, rt, εt),αrt+(1−α)pt)

]116


withPt(xt,yt,pt, rt, εt) = pt(β0+β1·pt+β2·max{pt−rt,0}+β3·min{pt−rt,0}+εt)−c(yt−xt)−G(yt,pt, rt)and the boundary conditionV *

T (xT , rT ) = 0 ∀xT , rT .(Note that G(yt,pt, rt) is defined as in Section 2.2with demand also depending on rt.)Reformulating equation (28) and (29) as in Feder-gruen and Heching (1999), such that V *

t (rt, xt) isreduced by cxt, results in

(30) V *t (xt, rt) = max

y≥x, pJt(yt,pt, rt)

Jt(yt,pt, rt) = (pt − γc)E[D(pt, rt)

](31)

+ −c(1 − γ)yt − G(yt,pt, rt)

+ γE[V *t+1(αrt + (1 − α)pt,yt − D(pt, rt, εt)

]and V *

T (rT , xT ) = cxT for all rT , xT .In the general theory of stochastic dynamic pro-gramming it is well known that a model gains incomplexity very quickly if the dimension of thestate space increases. In this way, both finding thecomputational as well as the analytic solution ofa model gets more complicated. However, if weassume joint concavity of the demand and revenuefunctions in all their variables, including the refer-ence price r, an adaptation of the proof in Feder-gruen and Heching (1999) still works and one canshow that a base-stock policy is optimal. The proofinductively shows that profit function Jt(y,p, r) isjointly concave in y, p and r and thus the opti-mal profit V *

t (x, r) is jointly concave in x and r (formore details see Gimpl-Heersink, Rudloff, Fleisch-mann, and Taudes (2007)). While Federgruen andHeching (1999) can also show the optimality ofa list-price policy, due to the additional complexityof the model in this section, a similar result has notbeen achieved so far. Unfortunately, the practicalusefulness of this base-stock result is question-able as the joint concavity assumption in all threevariables makes this proof unsuitable for manycommonly used demand functions like the linearone. In this case joint concavity of the revenuefunction does not hold any longer.However, since joint concavity in inventory level yand price p is sufficient for a base-stock policy,using the results of the one-period case, Gimpl-Heersink, Rudloff, Fleischmann, and Taudes(2007) establish the optimality of a base-stockpolicy for the two-period model with lineardemand.7 Changing the order of taking expected

7 A short sketch of the proof is given here. For a complete proofplease see Gimpl-Heersink, Rudloff, Fleischmann, and Taudes(2007).

values and summation in equation (29) we obtain

J1(x1,y1,p1, r1) = E[P1(x1,y1,p1, r1, ε1)(32)

+ γV2(x2(y1,p1, r1), r2(p1, r1), ε1)] .

It turns out that the profit of the second timeperiod V2(x2(y1,p1, r1), r2(p1, r1), ε1) is notnecessarily jointly concave for all x2. However,the joint concavity of the first time period’sprofit P1(x1,y1,p1, r1, ε1) is strong enough todominate the non-concavity of the future profit.This results in joint concavity of the profit functionJ1(x1,y1,p1, r1), leading to the optimality of thebase-stock policy.For the case of the linear demand model, one canstill find steady-state solutions. Assuming the lossneutral case β2 = β3, these are given by

(33) p* =(β1c − β0)(1 − αγ) + β2(1 − γ)c2β1(1 − αγ) + β2(1 − γ)

.

(34) y* = E[D(p*,p*, ε)] + F−1

(b − (1 − γ)c

h + b

).

Note that the optimal steady-state price p* in equa-tion (33) is the same as in Section 4.1 and equa-tion (34) corresponds to the optimal steady-statebase-stock level obtained in Section 3.2 for the op-timal steady-state price p* from equation (33). Theproof is a combination of the proofs for the steadystates of the models (7) and (24) from the abovesections. As in the proof of the joint pricing andinventory without reference effect case, we firstfind the steady-state inventory. Again the infinitesum representation of equation (28) is rearrangedas in the sections before to get

V (x1, r1)(35)

= cx1 +∞∑t=0

γt maxyt+1,pt+1

[pt+1E[D(pt+1, rt+1)]

− c(1 − γ)yt+1

+γcD(pt+1, rt+1) − G(yt+1,pt+1, rt+1)]

,

where

G(yt+1,pt+1, rt+1)(36)

= h

yt+1−E[D(pt+1,rt+1)]∫−∞

(yt+1 − E[D(pt+1, rt+1)] − u)f (u)du

+ b

∞∫yt+1−E[D(pt+1,rt+1]

(u − yt+1 − E[D(pt+1, rt+1)])f (u)du .

117


Figure 15: Optimal pricing decision (loss neutral)

Figure 16: Optimal ordering decision (loss neutral)

Differentiation with respect to y and using thesame arguments as in Section 3.2, we get thesteady state base-stock equation (34) dependingon the price. Note that the reference price hasto be equal to the price as this is necessary forthe steady-state price. The steady-state price informula (33) is found analogously to the proof ofthe steady state for the reference price model (24)after substitutingy*(p) fory in the infinite sum (35)and differentiating with respect to the price p.Interestingly, this yields exactly the same steady-state price p* as in Section 4.1.In the following we present the results for theloss neutral (loss averse) base example with β0 =100, β1 = −20, β2 = β3 = −40 (β2 = −50,β3 =−30), c = 0.5, α = 0.5, γ = 0.5, b = 0.4, h =0.005, T = 15 and a standard deviation σ = 20.Figures 15 to 18 show that a base-stock/list pricepolicy is optimal for all reference prices. Fromfigures 15 and 17 one can see that the optimalprice is always increasing in reference price, nomatter if we consider loss neutral or loss aversecustomer behavior. This is true in any time period.

Figure 17: Optimal pricing decision (loss averse)

Figure 18: Optimal ordering decision (loss averse)

However, the optimal ordering quantity is onlyincreasing in reference price in the loss neutralcase (see Figure 16). In contrast Figure 18 showsthat the optimal inventory after ordering is notmonotonous in reference price in the case of lossaversion. Due to this added complexity we willconcentrate on the loss neutral case for the rest ofthe paper.Similar to Figure 4, one can see from Figure 19 thatprice discounts for high inventory levels beforeordering are also given in the reference price case.However, due to the negative carry-over effects,these are generally smaller than in the classicalsetting. Also note the generally lower price level inthe model including reference price effects.In the model of Section 3.2 price discounts aregiven at a higher inventory level, the more time isleft in the planning horizon (see Figure 4). In themodel from this section price discounts are gener-ally smaller and are given as soon as the inventorylevel before ordering is higher than the base-stocklevel (see Figure 19). Since we give smaller dis-counts, we have to react earlier in time in order to

118


Figure 19: Price trajectories in inventory level x0

Figure 20: List-price trajectories in reference price

move the inventory level before ordering below thesteady-state level and thus reach a possible steadystate.The model in Section 3.2 aims at reducing base-stocks over time and thus list prices are increasingover time (see Federgruen and Heching (1999)). Ifvariance is independent of price, list prices tend tobe constant over time (see Figure 5). In contrastthe model in this section behaves qualitativelycompletely different. Here, similar to Figure 11,list prices are decreasing over time in order tobenefit from the reference price effects (see Fig-ure 20). With decreasing prices and also the result-ing reference price effect, expected demands areincreasing necessitating higher base-stock levels(see Figure 21).As in sections 2.2 and 4.1 one can study the be-havior of the steady-state solutions for our com-

Figure 21: Base-stock trajectories in reference price

bined model. Using the steady-state formulas (33)and (34) we can describe the behavior of thesteady states in the memory parameter α, thediscount factor γ and the reference effect |β2|.Note here that β2 ≤ 0. For examples of the be-havior with respect to these parameters see ta-ble 3. Differentiating p* with respect to α and β2,

we get ∂p*

∂α = γβ2(1−γ)(β0+β1c)(2β1(1−αγ)+β2(1−γ))2 < 0 and ∂p*

∂β2=

(1−γ)(1−αγ)(β0+β1c)(2β1(1−αγ)+β2(1−γ))2 > 0. Since we do not allow forprices below ordering costs, the inequalities holdas thus β0 + β1c > β0 + β1p > 0 for all p. Hence,

p* is decreasing in α and |β2|. As F−1(

b−(1−γ)ch+b

)is independent of both α and β2 and demand isdecreasing in price, E[D(p*, ε)] is increasing in αand |β2|. As a consequence, y* is increasing in αand |β2|. Furthermore, ∂p*

∂γ = −β2(1−α)(β0+β1c)(2β1(1−αγ)+β2(1−γ))2 > 0

and hence, p* is increasing in the discount factor.However, for γ one cannot make a definite state-ment about the base-stock, as the safety stock isincreasing in γ while E[D(p*, ε)] is decreasing in γ.As a result, in cases of small uncertainty in demand

Table 3: Steady-state base-stock andoptimal price

α 0 0.33 0.66 1

p*α 2.70 2.67 2.61 2.00

S*α 47 49 51 53

β2 0 −20 −40 −60

p*β2

2.75 2.65 2.55 2.47

S*β2

38 40 44 46

γ 0.75 0.85 0.95 1

p*γ 2.38 2.49 2.65 2.75

S*γ 62 67 76 90

119


Figure 22: Sequential optimization versus joint optimization

the base-stock is decreasing in γ whereas for largeuncertainty the base-stock is increasing in γ.For the rest of the paper we investigate the differ-ences between a sequential and a joint optimiza-tion approach (see Figure 22). In the sequentialapproach first the marketing/sales department de-termines an optimal price without considering anyinventory decisions and taking demand fulfillmentfor granted. This price is then passed on to the pro-duction unit of the company which then decideson an optimal stocking quantity without havingthe possibility to change the price. In the jointapproach both decisions are taken simultaneously.In figures 23 and 24 we compare the optimalbase-stock and price/reference price paths for the

Figure 23: Price path (sequential vs joint optimization)

sequential and joint approach from Figure 22. Forthe time being we assume r1 = p* and x1 = 0 toavoid having a transient phase at the beginningof the planning horizon. Using joint optimization,price and base-stock leave the steady state later.This is because we have the opposing strategies ofbenefitting from the reference effects by loweringprices towards the end of the planning horizon (seeFigure 11) and aiming at a clearance of stock at theend of the planning horizon (see Figure 5).The question now is how such a joint optimiza-tion of price and inventory increases the benefitsover the sequential optimization. Figure 25 showsthat similar to Figure 6 we get the largest benefitsof joint optimization towards the end of, or for

Figure 24: Inventory path (sequential vs joint optimization)

120


Figure 25: Profit increase over time

Figure 26: Profit increase in inventory level x0

a short planning horizon. In contrast to the com-parisons in Federgruen and Heching (1999), whobase all numerical results on a coefficient of varia-tion, we always use a constant standard deviation(not depending on price). As in Section 3.2 thisresults in generally relatively low benefits. AgainFigure 25 shows relatively low benefits, which canbe much higher for substantially larger inventorylevels before ordering (see Figure 26 for the lasttime period t = 50). Similar to Figure 7 this effectcan be observed for any time period t. However, incomparison to the model without reference effects,for smaller t this effect only appears for inventorylevels before ordering much higher than 200. Thisis because the pricing strategy under referenceprice effect enables us to clear higher stock levelsin later time periods.

Figure 27: Profit increase in reference effect

Figure 28: Profit increase in reference price r1

Figure 27 shows that the benefit of the joint modelwith reference effect is at least 10 times the ben-efit of the model without reference effect, and isconsiderably higher than when the reference effectgets larger. While in the classical setting price isonly varied to control inventory, here price has itsown dynamics, and incorporating the influence ofthe reference price increases the benefits of inte-grating pricing and inventory control significantly.Moreover, a significant difference to the model ofSection 3.2 is that while there the benefit convergesto zero for long planning horizons, here the benefitconverges to a value considerably higher than zero(depending on the parameters chosen). This effectis more prominent the more the starting referenceprice differs from the optimal steady-state price(see Figure 28).

121


5 Conclusion and Directions forFuture Research

In this paper we surveyed the results on the gainsof integrating pricing and inventory control forthe case when inventory can be set and demandis stochastic. Then we investigated whether usinga reference price model to describe demand in-creases the benefits of integration with a logisticmodel. Here, we studied whether the base-stocklist-price policy still holds in such a setting andinferred the size of potential benefits via numeri-cal simulation. We found that the benefits increaseconsiderably when reference effects are included,i.e. even with a constant standard deviation weachieved at least 10 times the benefit achieved bythe the joint model without a reference price. Also,it turns out that the base-stock policy also holdsfor the reference price model. However, except forthe two-period case with linear demand, this resultcannot be proved yet, but our conclusions are wellfounded on extensive numerical simulations. Wealso added details to existing literature by studyingsingle-period joint pricing/inventory control withbacklogging and by extending the steady-state ref-erence price proof from Popescu and Wu (2007) tothe case of positive ordering cost. We furthermoregive a steady-state price and inventory for the jointinventory and pricing models with and withoutreference effects.Clearly, we are only at the beginning of understand-ing combined reference price/inventory models:a general proof of the base-stock policy for lin-ear demand is needed; also, the dependence ofinventory after ordering and price on the statevariables needs to be studied. Fast algorithms forsolving (30) and (31) need to be developed, sameas suitable heuristics. As in practice the parame-ters of the demand model have to be estimated,combined estimation and control approaches area fruitful area of future research, too.

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BiographiesLisa Gimpl-Heersink is Diplom Ingenieur (Masterof Science) in Technical Mathematics (Operations Re-search, Statistics and Management Science) at GrazUniversity of Technology, Austria. Academic workingand teaching experience as assistant lecturer and sci-entific assistant at Graz University of Technology, andthe University of Leoben. Non-academic working expe-rience in life-insurance mathematics and logistic soft-ware algorithm design. Currently doctoral candidate in‘Economics and Social Sciences’ and research associatewithin the project ‘Integrated Demand and Supply ChainManagement’ at the Vienna University of Economics andBusiness Administration, Institute of Production Man-agement. Research interests: supply chain management,pricing and inventory control models (numerical and an-alytical optimization), dynamic programming.

Christian Rudloff is currently project assistant forthe WWTF project ‘‘Integrated Demand and SupplyChain Management’’ (WU Wien). MSc. and Ph.D. inMathematics, University of East Anglia, UK. StudiedMathematics with minor in Management, UniversityRegensburg, Germany. Experience in econometrics asproject assistant, FWF project ‘‘Identifikation multivari-ater dynamischer Systeme’’ (TU Wien). Industry experi-ence (CACI, London, 2002--2003) building geographicaldatabases for marketing management. Teaching expe-rience as a lecturer at Carnegie Mellon University andChatham College, Pittsburgh, USA (2003--2005). Re-search Areas: supply chain management, pricing andinventory control models, time series analysis, econo-metrics.

Moritz Fleischmann is an Associate Professor of Sup-ply Chain Management at Rotterdam School of Man-agement, Erasmus University. He received his Ph.D. inGeneral Management from Erasmus University in 2000.He was a visiting researcher at the Tuck School of Busi-ness and at INSEAD. Moritz is currently teaching in theMBA and MScBA programs of RSM Erasmus Univer-sity. His research addresses various topics in the field ofsupply chain management. Current focal points includecoordination of revenue management and operations,multi-channel distribution, in particular e-fulfillment,and closed-loop and sustainable supply chains. In thesefields, Moritz has been working with companies such asIBM, Heineken, and Albert Heijn.

Alfred Taudes Visiting Professor, University ofTsukuba, Japan (1997/98), Speaker of the SpecialResearch Area ‘‘Adaptive Information Systems andModeling in Economics and Management Science’’(2000--2005), Coordinator of the WWTF project‘‘Integrated Demand and Supply Chain Management.’’Within Call ‘‘Mathematics and ...’’ (2005--2008). Pub-lications, e.g. in Management Science, MIS Quarterly,Marketing Science, Journal of Management Informa-tion Systems, Professor for Business Administrationand MIS at the Vienna University of Economics andBusiness Administration, Institute for ProductionManagement. Head of Department for InformationSystems and Process Management. Research Areas:supply chain management, marketing science, revenuemanagement.

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