Coordinated Lot-Sizing and Dynamic Pricing under a Supplier All-Units Quantity Discount

We consider an economic order quantity model where the supplier offers an all-units quantity discount and a price sensitive customer demand. We compare a decentralized decision framework where selling price and replenishment policy are determined independently to simultaneous decision making. Constant and dynamic pricing are distinguished. We derive structural properties and develop algorithms that determine the optimal pricing and replenishment policy and show how quantity discounts not only influence the purchasing strategy but also the pricing policy. A sensitivity analysis indicates the impact of the fixed-holding cost ratio, the discount policy, and the customers' price sensitivity on the optimal decisions.


Introduction
In recent years, the coordination of marketing and operations decisions has received a lot of attention. However, in practice these two decision areas are usually still independent. Even when the performance of marketing and operations is independently optimized in order to achieve their respective best operating level, it may lead to suboptimal performance for the firm as a whole (Ghose and Mukhopadhyay, 1993). In particular, the coordination of pricing strategies and operations decisions is still at an early stage and offers significant opportunities for improving supply chain performance. In this context, dynamic pricing strategies use intertemporal price discrimination to achieve a better match of supply with demand. On the supply side, the order quantity is a function of customer demand and fixed and variable purchasing costs. On the sales side, demand depends on the charged selling price. Therefore, determining the optimal dynamic selling prices and optimal order quantities is interdependent. It is common in practice that marketing determines the selling price first without taking into consideration overhead â Corresponding author, Tel.: +49-621-181-1656+49-621-181- , Fax: +49-621-181-1653 purchasing or a supplier quantity discount. Then, given the selling price and the resulting demand forecast, operations decides on order quantities. However, due to the fact that marketing influences demand by the charged price and operations is responsible for satisfying this demand, price and procurement decisions are strongly connected and should be decided simultaneously. Fleischmann et al. (2004) review the linkage between pricing and operational decisions. They indicate different drivers for dynamic pricing strategies like promotions and other marketing-related instruments and more operations-driven pricing strategies known from supply chain coordination. It is a common practice in that suppliers offer a discount for large order quantities. For this kind of price discrimination, a supplier designs a menu of price-quantity pairs and customers select their optimal purchasing volume. Dolan (1987) and Wilcox et al. (1987) provide several reasons for a firm to offer quantity discounts from both a marketing and an operations management point of view. If the supplier faces high setup costs which lead to a large lot-size and high holding costs, quantity discounts may reduce the inventory level immediately after stocking due to larger customer orders. Furthermore, suppliers offer quantity discounts

BuR --Business Research Official Open Access Journal of VHB Verband der Hochschullehrer für Betriebswirtschaft e.V. Volume 1 | Issue 1 | May 2008 | 125--141
for a better utilization of idle capacity in order to achieve economies of scale in manufacturing. From a marketing perspective, quantity discounts are used to stimulate sales, e.g., Neslin et al. (1995). From a financial point of view, the time value of money is taken into consideration. Because of the offered quantity discount, buyers decide to buy earlier and a larger quantity. Therefore, revenues are available earlier for possible reinvestment (Beranek, 1967). In this paper we consider an economic order quantity (EOQ) model where the supplier offers an all-units quantity discount (AQD). This model is applicable to all goods that are purchased from a supplier that offers a quantity discount, e.g., durable consumer or industrial goods. We compare a decentralized decision framework where first marketing determines the optimal pricing strategy and then operations optimizes the replenishment policy to coordinated decision making where the retailer decides on pricing strategy and replenishment policy simultaneously. Hereby, we distinguish between two pricing strategies. In cases of constant pricing, the retailer determines the optimal selling price that is constant over an infinite planning horizon. In case of dynamic pricing the retailer varies the selling price over time. If price changes are free of charge, the optimal strategy is to change the price continuously over time (Rajan et al., 1992). If price changes are associated with costs, a continuous price adjustment is not beneficial. Therefore, there exists a finite optimal number of price changes. Netessine (2006) provides a variety of practical situations which cause that companies change prices only a limited number of times throughout the sales period. We analyze the benefits from coordinated dynamic pricing and replenishment compared to coordinated constant pricing and replenishment and to decentralized decision making. We assume that the number of price changes over an order cycle is given. However, in order to determine the optimal number of price changes, the results of Transchel and Minner (2005) can be used. They analyze a joint dynamic pricing and EOQ lot-sizing model without consideration of a supplier quantity discount and show that the optimal number of price changes and the optimal pricing and replenishment policy can be optimized by a two-stage program. That is, at the second stage, for a given number of price changes, the retailer determines the optimal pur-chasing and pricing strategy. At the first stage, the number of price changes is optimized anticipating the optimal purchasing and pricing strategy of the second stage. Transchel and Minner (2005) show how dynamic pricing can enhance operational efficiency by increasing the demand rate when inventories are high. The benefits of exploiting supply quantity discounts significantly depend on whether variable purchasing price reductions can offset additional holding costs from ordering minimum required quantities. Therefore it appears promising to use dynamic pricing here, too, to reduce the impact of holding costs. Furthermore, we extend Eliashberg and Steinberg (1993) who compare sequential and simultaneous optimization of lot-size and (constant) selling price without quantity discounts. They show that the optimal selling price in the case of simultaneous optimization is larger than in the case of sequential optimization which, in turn, results in a lower order frequency. This property does not necessarily hold if the supplier offers a quantity discount. We develop optimization models for three different decision frameworks: the decentralized framework where marketing and operations optimize independently, the coordinated-constant framework where the retailer optimizes a constant price and the order quantity simultaneously, and the coordinated-dynamic framework where the retailer employs a finite number of price changes over an order cycle. We provide analytical properties of the objective functions and present algorithms for determining the optimal replenishment policy and price strategy for the coordinated-constant and the coordinated-dynamic framework, respectively. The remainder of the paper is organized as follows. After a literature review in Section 2, we introduce the models and the underlying assumptions and properties in Section 3. Section 4 compares the three frameworks for the special case of a linear price-response function. In a numerical example presented in Section 5 we show the benefit of coordinated decision making and in particular dynamic pricing compared to a decentralized decision framework. Furthermore, we show the impact of changing fixed costs, discount policy, and price sensitivity on the optimal decisions and the resulting benefit of coordination. Finally, we conclude the results, discuss limitation of the model, and indicate directions for further research in Section 6.

Volume 1 | Issue 1 | May 2008 | 125--141
2 Literature review Several contributions analyze joint pricing and manufacturing situations. One of the earliest papers of integrated marketing/pricing and manufacturing decisions is by Whitin (1955) who shows that simultaneous decision-making on a constant price and ordering decision leads to significant profit increases compared to decentralized decisions. Eliashberg and Steinberg (1993) provide a comprehensive review of problems at the interface between marketing and operations. They compare differences between a decentralized and a coordinated optimization problem. In their overview, Bitran and Caldentey (2003) summarize dynamic pricing policies and their relation to revenue management. Elmaghraby and Keskinocak (2003) provide a review of literature and current practices in dynamic (intertemporal) pricing. Chan et al. (2004) review and classify papers according to a number of characteristics, e.g., stochastic vs. deterministic parameters and single vs. multiple products. Benton and Park (1996) and Munson and Rosenblatt (1998) provide literature reviews on quantity discounts where they classify the literature under several discount schemes, different perspectives (buyer, supplier, joint), and other criteria like planning horizon and number of products. The impact of quantity discounts on the economic ordering decision has been investigated in several settings. In the literature, the most common discount policies are the all-units and the incremental-units discount policies. When the supplier offers an allunits quantity discount (AQD), the reduced purchasing price applies to the entire order quantity once the order quantity reaches a critical breakpoint. An incremental-units discount, however, only applies to all units in excess of a particular breakpoint. Hadley and Whitin (1963) develop a procedure for determining the optimal economic order quantity for both all-units and incremental units discount schemes. This approach is included in almost every textbook on operations management and assumes that the demand rate is known and constant over an infinite planning horizon and that the decision maker follows the objective to minimize average costs. Gupta (1988) provides an improved procedure for determining the optimal lot-size by consid-ering an upper bound for the relevant cost and Goyal and Gupta (1990) propose a further simplification which requires only a few EOQ calculations for determining the optimal lot-size. In Abad (1988a) and Abad (1988b) the simultaneous optimization of lot-size and selling price when the supplier offers an all-units or an incremental units discount is analyzed. He develops a procedure for determining the optimal lotsize and the optimal selling price. Burewell et al. (1991) extend the model of Abad (1988a) and allow for planned inventory shortages. They derive a similar procedure as Abad (1988a) to determine the optimal lot-size and selling price for two classes of demand functions, iso-elastic and linear. More recently, discount pricing schedules have received growing attention to improve the coordination between vendors and buyers, see e.g., Weng (1995), Rubin and Benton (2003), and Wang (2005). Inventory problems with simultaneous optimization of the replenishment policy and a dynamic pricing strategy where only a limited number of price changes is allowed has been investigated less frequently. The main motivation for considering a limited number of price changes are organizational costs associated with each price change. Abad (1997) formulates a model where the reseller responds to a temporary price reduction of the supplier by an adjustment of the retailer's own selling price. Abad (1997) considers that the reseller is allowed to charge two selling prices in each order cycle. The presented model optimizes the first (discounted) selling price and fixes the second selling price to the optimal constant price. He presents a procedure where a temporarily reduced selling price yields a higher cycle profit than in the case of a static selling price. As shown in Transchel and Minner (2005), the majority of benefits from dynamic pricing can be captured by very few different price levels and, therefore, a discrete number of changes balances benefits and costs of price changes. Netessine (2006) analyzes a pricing problem with a limited number of price changes in a dynamic, deterministic environment and a capacity constraint. He characterizes the impact of capacity constraints on the optimal prices and the timing of price changes and provides several comparative statics results.

Assumptions
Consider a monopolistic retailer who is selling a product on a single market over an infinite planning horizon. We assume that customer demand follows a function of the selling price P and arrives continuously at a rate of DäPå which is a differentiable and strictly non-increasing function in P with DäPå OE 0 and D äPå < 0. Furthermore, let " define the price elasticity of DäPå with " ã á D äPå DäPå P which is the percentage change in demand in response to a percentage change in price. Based on the price elasticity, we define a class of price-response functions as follows: Defintion 1 A price-response function DäPå has an increasing price elasticity (IPE), if @" @P OE 0.
The intuition behind the IPE property is that with a price increase by a certain percentage demand decreases by a larger percentage. At every point in time, the demand rate depends only on the current price, i.e., customers are willing to buy as soon as the price is below their individual reservation price. LetP denote the market reservation price withP ‹ ‰. We assume that by charging a reduced selling price at the beginning of an order cycle customers do not behave strategically and buy in expectation of future prices (forward buying and postponement). We follow the assumptions of the EOQ model. With the release of any single order there is an associated setup cost F. Furthermore, the supplier has no capacity constraints and the overall order quantity is delivered in one shipment without any delay. Products delivered but not yet sold are kept in inventory subject to holding cost per unit and unit of time. The replenishment orders are placed in batches of size Q every T periods over an infinite planning horizon. Backorders are not permitted. The supplier offers a regular purchasing price c 0 per unit and an AQD schedule with l ã 0; · · · ; L different purchasing price categories where the discount is r l percent per unit if the order quantity is larger than or equal to a breakpoint quantityQ l . The AQD policy with multiple breakpoints is characterized by a vector èär 0 ;Q 0 å; är 1 ;Q 1 å; · · · ; är L ;Q L å | r 0 < r 1 < · · · < r L ;Q 0 <Q 1 < · · · <Q L é with r 0 ãQ 0 ã 0. Let c l ñã ä1 á r l åc 0 be the reduced purchase price for a unit if the order quantity Q ∈ aeQ l ;Q là1 å with c 0 > c 1 > · · · > c L and Q Là1 ã ‰. Inventory holding costs depend, among others, on the cost of capital that, in turn, depends on the purchase price c l and are denoted by h l per unit and unit of time. We assume that h l is an increasing function of c l .

Decentralized decisions
Assume that the selling price and the purchasing strategy are determined by separated decisionmaking units. First, marketing optimizes the selling price and generates customer demand. Then, given this demand forecast, operations determines the optimal replenishment policy taking into account the supplier's quantity discount. Marketing does not take into account fixed ordering costs and does not anticipate purchasing price discounts as a result of order quantities because the discount that is actually applied is unknown until the operations decision is taken. Marketing's objective function is as follows: (1)ΠäPå ã äP á c 0 åDäPå : The optimal selling priceP â is obtained from the first-order condition P à DäPå D äPå ã c 0 . Given the demand forecastD â ã DäP â å, operations minimizes average costs of replenishment and inventory taking into account the supplier quantity discount. Hadley and Whitin (1963) developed a two-stage algorithm in order to determine the optimal replenishment policy. A first stage iteratively calculates the constrained economic order quantityQ â l and the resulting costs starting from the highest discount c L until the first index l 0 is found where the solution satisfiesQ â l 0 OEQ l 0 andQ â l <Q l for all l > l 0 . Thus, At a second stage, the cost of this inner solutionC â l 0 is compared with the costs of all breakpoint quantities larger thanQ â l 0 , i.e.,C l äQ l å for l ã l 0 à 1; :::; L withC

Coordinated-constant decision
In case of coordinated decision making, we simultaneously optimize lot-size Q and selling price P.
The optimization problem for a particular purchasing price c l is given by A relaxation of (4) and differentiating (3) with respect to Q and P yields the necessary first-order conditions for an inner solution (e.g., see Whitin (1955) and Eliashberg and Steinberg (1993) where the optimal selling price P â l , conditional that the purchasing price is c l , is represented as an implicit function of the optimal order quantity. For a particular purchasing price c l the optimal order quantity is only feasible ifQ l ‹ Q â l . Given this constraint, a transformation of Q â l from (5) yields that an inner solution P â l is only feasible if P â l is lower than a break priceP l for a given unit purchasing cost c l : where D á1 ä:å denotes the inverse of the priceresponse function. The inverse function does exist due to the fact that DäPå is a strictly monotone function. Using this result, the AQD policy can be characterized by èär 0 ;P 0 å; är 1 ;P 1 å; · · · ; är L ;P L å | r 0 < r 1 < · · · < r L ;P 0 >P 1 > · · · >P L é whereP 0 ãP. Substituting the optimal order quantity into (3), the two-variable problem is reduced to a single-variable problem that only depends on P: Π l äPå ã äP á c l åDäPå á 2Fh l DäPå ; s:t: P ‹P l :

Properties of an optimal pricing and lotsizing policy
The following properties characterize the profit function (7). (7) is either a concaveconvex function or a strictly concave function of P with lim P→0 Π l äPå < 0 and lim P→P Π l äPå ã 0.
The proof of Property 1 is given in Appendix A.1. Abad (1988a) reduces (3) to a single-variable problem that only depends on the order quantity Q. He shows that if the first-order condition with respect to P yields a closed-form solution P â äQå, as it is the case for linear and iso-elastic price-response functions, the reduced profit function is a convexconcave function of Q and develops a procedure to determine the optimal price and lot-size. Based on this result, Abad (1988a) gives an algorithm to determine the overall optimal lot-size. Note that we do not need the required condition of having a closed-form solution for P â äQå.
Property 2 For an arbitrary fixed selling price P, (7) is a strictly decreasing function of c l . Therefore, the profit functions Π l and Π l for different unit purchasing costs c l and c l do not intersect and Π l äPå > Π l äPå for all P and c l < c l .
From the partial derivative of (7) with respect to c l and the assumption that h l increases in c l , it is easy to verify that @Π l @c l < 0. Implicitly, it follows that the profit function Π l äPå > Π l äPå for c l < c l .
Theorem 1 Let l 0 be the largest index of a discount where the local optimum Q â l 0 is feasible, i.e., Q l 0 ‹ Q â l 0 andQ l > Q â l for all l ã l 0 à 1; · · · ; L. Then for all l < l 0 , Π l äPå < Π l 0 äP â l 0 å for all P. The proof follows directly from Properties 1 and 2 with the implication that all discounts that are lower than the discount r l 0 can be omitted for the determination of the optimal solution. If Π l äP â l å < 0 for all c l , then Q â ã 0. To find the optimal value of P, we have to find the profit maximizing price in each interval äP là1 ;P l ç and compare these profits to determine the global optimum. For the following illustration we assume that the supplier offers a single price breakQ 1 . Then, there are 3 cases for the optimal price P â .
1. The free local optimum P â 1 for the reduced purchasing price c 1 is a feasible solution, i.e., P â 1 ‹P 1 .

BuR --Business Research Official Open Access Journal of VHB
The free local optimum P â 1 is infeasible and the breakpoint profit (the retailer orders the breakpoint quantityQ 1 at the reduced purchasing price c 1 ) is larger than the optimal profit given the regular purchasing price c 0 , 3. The free local optimum P â 1 is infeasible and the profit where the retailer orders the breakpoint quantity is lower than the optimal profit for the regular purchasing price, Π 1 äP 1 å < Π 0 äP â 0 å.
In Case 1, as shown in Figure 1, the breakpoint priceP 1 is large enough such that the optimal selling price P â 1 is a feasible solution. Figures 2 and 3 illustrate Cases 2 and 3 where P â 1 is an infeasible solution. In Case 2 the breakpoint profit is larger than the optimal profit for the regular purchasing price c 0 and in Case 3 the opposite holds. If Π 1 äP 1 å ã Π 0 äP â 0 å, the retailer is indifferent between orderingQ 1 or Q â 0 . The bold lines represent the feasible profit curves. Properties 1 and 2 and Theorem 1 allow the following algorithm for calculating the optimal selling price.

Algorithm 1 for determining the optimal price and lot-size
Set l ñã L,

REPEAT
Calculate the optimal selling price P â l by solving the first-order condition @Π l @P ð ã 0, calculate Π â l ñã Π l äP l å, l ñã l á 1; UNTIL P â l <P l . (The loop stops at the index l 0 , the largest index for which P â l 0 ‹P l 0 , i.e., the local optimum P â l 0 is an inner solution.) Calculate Π â l 0 ñã Π l 0 äP â l 0 å. The optimal profit is determined from Let l â ñã argmaxèΠ â l é. Then the optimal price and the optimal order quantity are given by

BuR --Business Research Official Open Access Journal of VHB Verband der Hochschullehrer für Betriebswirtschaft e.V. Volume 1 | Issue 1 | May 2008 | 125--141
The important property for this algorithm to work is that the profit curves do not intersect which gives that the profit function for a discount rate r l is located above of profit functions for discount rates r l with l < l.

Impact of coordinated decision making on price and purchasing policy
Now we compare decentralized (sequential) and coordinated (simultaneous) decision making on pricing and purchasing strategy. Eliashberg and Steinberg (1993) analyze the sequential and coordinated decision-making problem without a supplier quantity discount. They show that decentralized decision making, where marketing first decides on the selling price without taking into account operation's overhead costs and then operations determines the optimal purchasing policy, leads to an underestimated selling price and an overestimated demand rate. This, in turn, leads to larger lot-sizes than in case of coordinated decision making. However, if the supplier offers a quantity discount, these effects do not necessarily hold anymore. For a particular discount rate r l , a comparison of the first-order condition of decentralized and coordinated decision making gives DäPå D äPå àP ã c 0 and (9) DäPå D äPå à P ã c l à F Q for all l ã 0; · · · ; L : Under decentralized decision making, the optimal selling priceP â is only based on the regular purchasing costs c 0 . However, in the coordinated case, the price P â takes into account both overhead costs and an eventually used quantity discount.

Property 3 If
DäPå has an increasing price elasticity and the retailer does not use a quantity discount äl ã 0å, then the optimal selling price in case of decentralized decision making is lower than under simultaneous optimization, i.e.,P â < P â and DäP â å > DäP â å. If l > 0, the coordinated case yields a higher price in comparison to the decentralized system if the optimal coordinated selling price satisfies P â l > D á1 Fh l 2äc 0 ác l å 2 .
The proof is given in Appendix A.2. The direction of the price change is influenced by two effects: the overhead cost effect and the discount effect. The overhead cost effect describes the impact of coor-dinated decision making. In the case of sequential decision making, marketing does not take into account that operations uses a quantity discount. Depending on the actually paid purchasing costs, marketing either underestimates or overestimates operations costs that additionally contain setup cost and holding costs. The discount effect leads to a decreasing selling price. Due to the increasing lot-size by using the supplier quantity discount, the retailer decreases the prices in order to increase the demand and reduce holding costs. Whether the optimal selling price increase or decrease compared to the decentralized case depends on which effect dominates.

Coordinated-dynamic decision
Let us assume that the retailer can change the selling price N times at time t i ; i ã 0; · · · ; N á 1 within each order cycle where each price change is subject to administrative costs. äN å defines a non-decreasing function of N. At time t 0 ã 0, the retailer places the order and sets the selling price P 1 which holds until t 1 . At time t 1 , the retailer changes the price to P 2 . We assume that over a time interval aet iá1 ; t i å, the retailer charges a price P i . Let t N denote the cycle length. The objective is to optimize the purchasing and dynamic pricing strategy which includes the determination of the optimal prices and the optimal times where the price is adjusted. This problem can be solved by a two-stage optimization approach: At the second stage, for a given N, the retailer determines the optimal purchasing and pricing strategy. At the first stage, the optimal number of price changes N is optimized anticipating the optimal purchasing and pricing strategy of the second stage. For a simplified notation, let D i ã DäP i å. The two-stage problem can be formulated as follows: Equation (11) represents the average retailer profit given c l , a price vector P N ã äP 1 ; · · · ; P N å and a timing vector t N ã ät 1 ; · · · ; t N å. The first term is the unit revenue minus direct purchasing cost (when the retailer uses the discount rate r l ) for each interval multiplied by the respective demand rate and the length of the interval. In the second and third term we subtract inventory holding costs. These consist (see Figure 5) of the average of initial and final inventory in each interval (triangles) and the inventory that has to be carried over the entire interval aet i á t iá1 å to cover the demand of all subsequent intervals j ã i à 1; · · · ; N with an amount of N jãià1 D j ät j á t já1 å (rectangles). Constraint (12) ensures that the selling price does not exceed the reservation price and by constraint (13) the lot-size must be larger than or equal to the breakpointQ l to ensure that the used unit price is attained. A detailed analysis of how to solve (11) is given in Transchel and Minner (2005). The order quantity can be determined as the sum over all intervals, demand rates multiplied by the interval length and therefore implicitly by the price decisions in all subintervals of an order cycle

Properties of an optimal pricing and lotsizing policy
By algebraic manipulations (see Appendix A.3), we can simplify (11) to In order to maximize the average profit, we differentiate (15) with respect to P i and t i , i ã 1; · · · ; N. The necessary first-order conditions with respect to P i give The optimal price P â i in a particular time interval aet iá1 ; t i å for i ã 1; · · · ; N must satisfy (16) and depends on the initial and final point of this interval. For price-response functions with a price-elasticity that is non-decreasing in price, the optimal selling price P â i is non-decreasing in i (see Transchel and Minner (2005)). Figures 4 and 5 illustrate the optimal pricing policy and the corresponding inventory development for a certain N and c l . The first-order condition of Π äN å l with respect to t i for i ã 1; : : : ; N á 1 gives

BuR --Business Research Official Open Access Journal of VHB Verband der Hochschullehrer für Betriebswirtschaft e.V. Volume 1 | Issue 1 | May 2008 | 125--141
Therefore, the optimal times t â i must satisfy i ã 1; · · · ; N á 1: From (17) we find that the optimal time t â i is a function of the optimal price P â i of the current interval, the optimal price P â ià1 of the next interval, and the corresponding optimal demand rates. Given the results of (16) and (17), it is apparent that t â i can be represented as a function of its predecessor t â iá1 and its successor t â ià1 . With the initial condition t 0 ã 0 it follows that t â 1 is only a function of its successor t â 2 . Furthermore, t â 2 is a function of t â 1 and t â 3 . By substituting t â 1 as a function of t â 2 and some algebraic manipulations, the time t â 2 can be represented as a single-variable function of t â 3 . An iterative transformation and substitution gives that from aet â 1 ät 2 å, t â 2 ät 1 ; t 3 å, · · · ; t â N ät Ná1 åç follows aet â 1 ät 2 å, t â 2 ät 3 å, · · · ; t â Ná1 ät N åç. Now, by backward insertion, each optimal time where the selling price is changed can be represented as a function t â i ät N å of the cycle length. Therefore, the optimal prices and the optimal demand rates can be reduced to functions that only depend on the cycle length t N , e.g., äP â i ät N å; D â i ät N åå, and the maximization problem reduces to the following single-variable problem: In Section 4, we will show that for a linear priceresponse function this iterative procedure yields closed-form expression for P â i ät N å for all i ã 1; · · · ; N. In the general case, the dependencies have to be solved numerically. As for the static pricing problem the profit function Π äN å l ät N å has to possess some particular characteristics that enable to find the optimal solution efficiently.
For general price-response functions the proof of Conjecture 1 is analytically intractable. However, in the next section we show for linear price-response functions that Conjecture 1 holds. The shape of (18) depends on the price-response function. If DäPå is such that (18) is concave, quasi-concave, or concave-convex, similar to the constant pricing case, for a fixed c l , the following algorithm can be applied to determine the optimal solution. Otherwise a complete enumeration over all inner solutions and breakpoints has to be used in order to determine the optimal solution. Similar to the results under static pricing, a simple algorithm can be used to determine the optimal cycle t N for any c l .

Algorithm 2 to determine the optimal price strategy and lot-size
Set l ñã L,
The optimal profit is determined from

BuR --Business Research Official Open Access Journal of VHB Verband der Hochschullehrer für Betriebswirtschaft e.V. Volume 1 | Issue 1 | May 2008 | 125--141
Let l â ã argmaxèΠ äN å â l é. For the case Π äN å â > 0, the optimal cycle lengtht â N is either t â N if l â ã l 0 or t N l if l â > l 0 and the optimal order quantity is determined by Otherwise, it is not optimal to order, i.e., Q â ã 0.

END
This procedure resembles the algorithm for the classical EOQ average cost minimization problem with an all-units quantity discount. It takes into account the interdependencies of the determination of the optimal order quantity, the optimal prices in the N time intervals, and the optimal times where the price is adjusted. In Step 1 we solve the constrained optimization problems backwards, starting with the highest discount rate r L until an unconstrained solution is feasible for the first time. In Step 2, we compare the various breakpoint profits for L to l 0 à 1 to the first free optimal solution and identify the optimal discount rate r â l where the profit reaches the maximum. In Step 3 we determine the resulting optimal cycle length and the optimal order quantity.

Linear price-response function
In this section, we detail the previous results for the special case of a linear price-response function. We show that for each discount rate r l both the optimal prices and the average profit can be uniquely expressed as a closed-form single-variable function of t N , e.g., P â i ät N å for all i ã 1; · · · ; N and Π l ät N å. The linear price-response function satisfies all conditions of Section 3.1. We will show that Π l ät N å is a concave-convex function of t N and provide a detailed algorithm for determining the overall optimal decisions of the dynamic coordinated selling price strategy and the lot-size. A detailed analysis of the decentralized framework and the coordinated-constant pricing framework for linear price-response functions is given in Eliashberg and Steinberg (1993) and Abad (1988a). Let where a represents the market potential and the reservation price isP ã a b . Using the results of Transchel and Minner (2005), the optimal times t â i can be represented by func- the optimal prices and demand rates are characterized by and by inserting (22), (23), and (24) into (18), the profit function for a given purchase price c l can be transformed into the single-variable function For a detailed derivation of (25) see Appendix A.4.

Property 5
If D ã DäPå is linear, the profit function Π äN å l ät N å is a concave-convex function of t N .
The proofs of Property 4 and 5 are given in Appendix A.5 and A.6. With the results of Properties 4 and 5, all requirements to use Algorithm 2 are satisfied.

Algorithm:
Step (1): Solve the constrained optimization problem (19)--(20) from l ã L until the largest index l 0 is identified whereQ l ‹ Q â l 0 <Q l 0 à1 by using the Lagrangian Multiplier method. We introduce the Lagrangian multipliers 1 and i 2 for i ã 1; · · · ; N related to the minimum order quantity constraint and the reservation price constraints for all charged selling prices over an order cycle. The Lagrangian

BuR --Business Research Official Open Access Journal of VHB Verband der Hochschullehrer für Betriebswirtschaft e.V. Volume 1 | Issue 1 | May 2008 | 125--141
function L l for a particular purchasing cost c l is defined by: with the Kuhn-Tucker conditions We seek the price vector P â 1 ; P â 2 ; · · · ; P â N and the cycle length t â N that satisfy all these conditions. The partial derivation with respect to P i for all i ã 1; · · · ; N; and the partial derivation with respect to t N gives Subject to the complementary slackness condition, we obtain the first-order condition The two constraints (28) and 1 OE 0 are complementary inequalities, i.e., @L l @ 1 > 0 if and only if 1 ã 0. Thus, if 1 > 0, it must be satisfied that @L l @ 1 ã 0. The value of 1 can be interpreted as shadow price that values a decrease of the breakpoint quantityQ l by one unit. Equation (26) indicates that if h l > 0 the optimal selling price is strictly increasing over the order cycle. Therefore, the reservation price constraints must not be binding for i ã 1; · · · ; N á 1, that is, i 2 ã 0 for i ã 1; · · · ; N á 1. The only reservation price that may constrain the problem is for P N ät N å ‹P N : A similar shadow price interpretation holds for N 2 . Here, (29) and N 2 OE 0 are complementary inequalities, i.e., N 2 > 0 if and only if P N ã a b . Substituting P N from (26) into (29) gives an upper bound for the optimal cycle length For determining the optimal solution äP â 1 ; · · · ; P â N ; t â N ; â 1 ; N â 2 ), we solve the system of equations and inequalities (26), (27), (28), and (30) under the condition â 1 OE 0 and N â 2 OE 0.
Step (2) and Step (3) follow straight just as described in Algorithm 2.

BuR --Business Research Official Open Access Journal of VHB Verband der Hochschullehrer für Betriebswirtschaft e.V. Volume 1 | Issue 1 | May 2008 | 125--141
structure DäPå ã a á bP. The parameters are set to a ñã 1000, c 0 ñã 5, and h l ñã 0:02c l . The parameters F,Q 1 , r 1 , and b are varied in order to show the impact on lot-sizing and pricing. In the case of a dynamic pricing strategy, we assume that the retailer changes the selling price twice in each order cycle, i.e., N ã 2.
As a base case, we consider that the price sensitivity factor is b ã 100 and the supplier discount is r 1 ã 15ô if Q OE 4000 ãñQ 1 units. Furthermore, we distinguish F ã 0 and F ã 500. Due to the fact that purchasing price and interest rate are fixed, F ã 0 represents the case of a low and F ã 500 the case of a high fixed-holding cost ratio. Table 1 shows the optimal price and lot-sizing decision as well as the optimal cycle length and the optimal average profit for the three frameworks and both purchasing options regular äl ã 0å and discounted äl ã 1å. Column``Loss'' illustrates the loss of profit compared to the coordinated-dynamic framework. If F ã 500 and the retailer does not use the quantity discount (l ã 0), we verify the findings of Eliashberg and Steinberg (1993) that decentralized decision making leads to an underestimation of price and an overestimation of lot-size compared to coordinated decision making ä7:5 < 7:7 and 1414 > 1361å. However, this is not neces-sarily true if the supplier offers a quantity discount. If the retailer uses the quantity discount äl ã 1å, for both F ã 0 and F ã 500 it can be observed that the optimal selling price decreases between the decentralized and coordinated-constant framework ä7:5 > 7:1å whereas the order-quantity is equal. In case of decentralized decision and F ã 0, the optimal purchasing strategy is a just-intime strategy with the regular purchasing cost c 0 äΠ â 0 ã 625 > 600 ã Π â 1 å and an optimal selling price P â 0 ã 7:5. However, if the retailer optimizes price and purchasing policy simultaneously, the discount strategy becomes more profitable. That is, the discount effect dominates the coordination effect (see Section 3). It is optimal to accept additional holding costs in order to save variable purchasing costs. Due to the larger lot-size which would result in higher holding costs, it becomes more beneficial to decrease the selling pricing in order to increase the demand rate. In case of F ã 500, the lot-sizing decision is equal for the decentralized and coordinated framework. However, in the coordinated framework a lower selling price leads to a higher order frequency (a higher demand rate leads to a lower cycle length) which results in reduced holding cost. Furthermore, it can be seen that the benefit of dynamic pricing   (2:9,5:9) (7:6,7:7) 452 dynamic 1 4000 (7:6,13:4) (6:9,7:1) 713 4000 (6:8,13:6) (6:9,7:3) 673  compared to coordinated constant pricing for both F ã 0 and F ã 500 is lower than 1ô. The same observation can be made when we compare the dynamic-coordinated framework with the decentralized framework for the case F ã 0. If F ã 0, the benefit of coordinated-dynamic pricing increases to 4ô per time unit. Tables 2 and 3 illustrate the impact of the discount policy on the optimal solution. Table 2 shows the results of a decreasing minimum order quantity and Table 3 the results of an increasing discount rate. Both changes yield an increasing desirability of the quantity discount. While the benefit of the coordinated-dynamic framework compared to the coordinated-constant framework remains low, the benefit compared to the decentralized framework does increase. Compared to the base case F ã 0 where a JIT policy was optimal, now the quantity discount is beneficial such that the retailer orders the minimum order quantity. Table 4 illustrates the impact of price sensitivity on the optimal decision and the average profit. Compared to the base case, an increasing price sensitivity leads to a decreasing selling price among all decision frameworks. For F ã 0, across all decision frameworks, it is optimal to follow a JIT strategy. Therefore, decentralized decision making yields the same performance as coordinated decision making. But, in case of F ã 500, the application of coordinated-dynamic pricing is highly beneficial compared to decentralized and coordinated-constant decision making.

Conclusion
This paper considered an economic order quantity (EOQ) model when the supplier offers an allunits quantity discount (AQD) and the retailer faces a price-dependent demand rate. We analyzed three different decision frameworks, a decentralized decision-making strategy, a coordinated strategy with a constant selling price, and a coordinated strategy where the retailer is allowed to implement a finite number of price adjustments within each order cycle. For the coordinated frameworks, we derived analytical properties used to develop efficient algorithms in order to determine the optimal price and lot-sizing strategy. These algorithms are applicable to all concave, quasiconcave, and concave-convex profit functions. In a numerical example we showed the difference in decision making between the decentralized, the coordinated-constant, and the coordinated- dynamic framework. It could be observed that in the case of low price sensitivity of the customers, the benefit of dynamic pricing compared to constant pricing is rather low. However, with increasing price sensitivity, coordinated decision making and in particular dynamic pricing becomes more beneficial. Without a supplier quantity discount, the coordinated framework with a constant pricing strategy yields a higher price, a lower demand rate, and a lower order quantity compared to sequential pricing and replenishment (see Eliashberg and Steinberg (1993)). If the supplier offers an allunits quantity discount, these properties do not necessarily hold. In the discount case, two effects influence the outcome of decision making. The first effect is the overhead cost effect. In contrast to decentralized decision-making, a central decision maker takes into account all relevant costs (setup cost and inventory holding cost). This yields an increasing selling price, a decreasing demand rate, and a decreasing lot-size. The second effect is the discount effect. Here, under decentralized decision making, marketing does not consider the quantity discount schedule of the supplier and optimizes the selling price only using the undiscounted purchasing price. Thus, marketing overestimates the cost which yields a higher selling price than in the case of coordinated decision making. However, by taking into account the supplier quantity discount it might be optimal to order a larger lot-size at a reduced purchasing price. Therefore, an increase or decrease of the selling price, the demand rate, and the order quantity depends on which effect dominates, the overhead cost effect that yields an underestimation of the average cost in the decentralized system, or the quantity discount effect where the decentralized system overestimates the average cost. A weakness of the prescribed model is the deterministic and stationary environment based on the EOQ assumptions. Furthermore, we disregard strategic behavior of customers. Additional research is needed to analyze the problem where customers are willing to buy earlier (forward buying) or later (postponement) if prices vary over time. A natural extension of the model would be to consider a joint dynamic pricing and lot-sizing determination problem where the supplier offers an incremental units discount. An extension regarding supply chain coordination is to investigate the problem, how does the supplier design a quantity discount schedule if it is anticipated that the retailer employs decentralized decision making, coordinated decision making with a constant price, or coordinated decision making with a dynamic pricing framework.
However, it is not assured that c 0 < c l à F Q â l for a quantity discount r l with l > 0. From the fact that @ @P DäPå D äPå à P OE 0, it follows that the selling price in the coordinated system is higher if and only if The property follows from c 0 á c l < Fh l 2DäPå ⇔ DäPå < Fh l 2äc 0 á c l å 2 ⇔ P â l > D á1 Fh l 2äc 0 á c l å 2 :

A.3 Derivation of equation (15)
An expansion of the term Therefore, (11) can be modified to