Dynamic Pricing through Customer Discounts for Optimizing Multi-Class Customers Demand Fulfillment

Transcription

1 Dynamic Pricing through Customer Discounts for Optimizing ulti-class Customers Deman Fulfillment Qing Ding Panos Kouvelis an Joseph ilner# John. Olin School of Business Washington University St. Louis, O 6330 ingq@olin.wustl.eu Kouvelis@olin.wustl.eu #Joseph L. Rotman School of anagement University of Toronto 05 St. George Street Toronto, ON 4R N5 milner@rotman.utoronto.ca arch 2003

2 ABSTRACT We are concerne with the tactical problem of allocating inventory over a horizon to eman from several classes of customers when partial backlogging of unfulfille eman is possible. The customers are istinguishe into several classes by the contractual price they are to pay for the item. Deman from each customer class is moele as a realization of a (non-stationary) ranom variable uring each of several stages a perio is ivie into. The firm is able to view this eman in each stage prior to making an allocation ecision on which eman to fill. Unfille eman may then be partially backlogge for fulfillment at the later stages or the beginning of the next perio. The probability of this occurring is influence by a iscount the firm may offer as well as some class specific parameters. We present a solution approach to the problem of etermining the inventory allocation, the customer iscounts an the prioritization of eman for all stages (referre to as the ADP problem), through ynamic programming starting first with the final stage an then solving the problem by inuction. We show that a class orer policy is optimal with waiting customers being serve subsequent to new eman in any stage. We show how to fin the inventory allocation in any stage for each class, with the use of class specific threshol limits an show that these limits are monotonic in the waiting eman an the inventory remaining. We also iscuss the optimal etermination of offere iscounts. As the number of stages in a perio increases to infinity, the ADP solutions converge to a limit, which provies a theoretical optimal solution for the continuous-time eman cases. For the continuous time eman case, we provie an efficient an robust heuristic for its solution in real time. Our numerical results clearly inicate the avantageous shifting of the inventoryservice frontier through the implementation of ADP policies. The ADP solution always increases the expecte profit vis-à-vis the FCFS or no price iscounting policies, often in the range of 5-20%. Through appropriate iscounting that leas to significant customer retention an increases eman waiting rates of price sensitive customers, ADP policies reuce the base stock levels as well as the incurre holing an congestion costs in almost all cases while increasing the total fill rate of all classes an the prompt fill rates of the most profitable, but time sensitive customers.

3 . INTRODUCTION In this paper we stuy the tactical problem of allocating inventory over a perio to eman from several classes of customers when partial backlogging of unfille eman is possible. The customer classes are istinguishe by the contractual price they are to pay for the item an their willingness to wait for fulfillment of eman in a subsequent perio. Deman from each customer class is moele as a realization of a (nonstationary) ranom variable uring each of several stages a perio is ivie into. The firm is able to view this eman in each stage prior to making an allocation ecision on which eman to fill. Unfille eman may then be backlogge for fulfillment. The probability of this occurring is influence by a iscount the firm may offer as well as some class specific parameters. The problem arises in a number of inustries. The motivating example is base on the fulfillment of eman at a wholesaler of inustrial proucts. At the firm's istribution center, orers are receive throughout the ay from customers for whom there is a fixe price for a unit (generally from contractual terms) an who expect same ay shipping. Given a limite inventory, the istributor may choose to offer the customer next ay shipping on the item in hope of being able to fulfill the request of a more value customer. In orer to inuce the customer to wait for supply, the istributor may offer a iscount. Similar problems are foun in on-line catalog businesses where firms nee to etermine their availability to ship on a given ay. For example, an on-line bookseller may quote a time until shipping that is base on the ability of the firm to withraw a unit of eman from a warehouse. Customers arriving early in the ay may be quote a longer time until shipping so that inventory may be reserve for customers coming later in the ay with higher value orers. In this paper we show how to etermine the inventory allocation, iscounts to offer an customer-class prioritization (referre to as the ADP problem) in a moel where each perio is ivie into a number of stages. We show that inventory shoul be allocate in each stage in class-orer as long as the inventory is above a etermine threshol level. These threshol levels are shown to be monotonic in the waiting eman an the inventory remaining. We also show how to etermine the optimal iscounts to customers that are enie inventory. The initial inventory level is etermine by a base-stock orering policy. As the number of stages in a perio increases to infinity, the ADP solutions converge to a limit, which provies a theoretical optimal solution for the continuous-time case. The paper contributes to the literature by incorporating pricing 2

4 (through iscounting) with inventory rationing in a moel which can be solve in a relatively efficient manner. The research is relate to work in inventory rationing, yiel management an ynamic pricing. Early work by Topkis (968) consiere the rationing of inventory to eman from n customer classes when a perio is ivie into several intervals. He shows that a base-stock orering policy is optimal an eman is fulfille in class orer as long as inventory is above a class-epenent allocation level. These levels are foun for the no-backlogging case or full backlogging case. In our paper, we consier the partial backlogging case where price iscounts can be assigne to inuce backlogging. Cohen, Kleinorfer an Lee consier an (s, S) inventory system where two classes of customers arrive, with the higher priority customer being serve first. The focus of the paper is on the etermination of the reorer level s an the orer-up-to value S through the evelopment of heuristics an approximations. Ha (997, 2000) consiers the problem of allocating inventory to n customer classes in a make-to-stock environment where stock replenishment is explicitly moele as a prouction system through a /E k / queue. The optimal policy is characterize by an inventory level below which prouction is initiate an an inventory level for each customer class above which eman will be fulfille for the class. All enie eman is lost. Gerchak, Parlar an Yee (985) consier a two concurrently-arriving class problem where the ecision is when to reect lower class customers base on the time to go an inventory. Weatherfor, Boily an Pfeifer (993) stuy the etermination of ynamic allocation limits in a two-class problem with the possibility of lower-class customers purchasing at a higher class price. Another relate stream of work in the inventory management literature looks on economic incentives to retain customers in the presence of stockouts. oinzael an Ingene (993) consier the profit maximizing strategy of a istributor that hols two ifferent but substitutable goos: One for immeiate an one for elaye elivery. Outof-stock situation can lea to partial eman loss, as some of the customers might wait for elivery of their goo or switch to the available one. It is shown that a profit maximizing strategy may entail setting a price for the elaye elivery so as to encourage switching behavior. The main ifferences with our work are the lack of inventory rationing consierations an the ability to ynamically aust prices offere to backorers customers. Cheung (998) consiers a continuous review moel where a iscount can be offere to the customers willing to accept backorers even before the inventory is eplete, but the proportion of backorering customers is not a function of monetary incentives, as is the case in our work. The optimality of offering backorering incentives for a simple inventory system is explore in DeCroix an Arreola Risa (998), but their 3

5 analysis oes not exploit ifferent customer classes or ynamic iscount austments. Further, their analysis seems to imply that the maority of cost savings is as a result of offere backorere incentives after the stockout occurs, which is contraicte in a multiclass profit ifferentiate customer eman environment. The problem is also relate to the well-stuie yiel management problem in which a firm seeks to etermine the number of units of capacity to reserve for sale to customers arriving at a later time. A lot of that work was motivate by the airline inustry practices, an sets prices for flights in the presence of multiple customer classes. Initial work on the problem for a single flight was consiere by Littlewoo (972) for the case of two fareclasses. Belobaba (987) consiere a heuristic approach to solving the multiple fareclass problem. Wollmer (992), Brumelle an cgill (993) an Robinson (995), consiere extensions an refinements etermining optimal solutions. In all these moels fare classes are assume to arrive sequentially so that the solution of the problem require etermining how much to reserve for other fare-classes. In our paper, we assume concurrent arrivals of eman from ifferent classes. A review of the extensive yiel management literature is provie by cgill an van Ryzin (999). A number of recent papers stuy a ynamic version of the perishable asset revenue management problem where price may be varie continuously over time. Gallego an van Ryzin (994) consier a moel where the eman rate for an item epens on the current price offere an solve for an expecte revenue maximizing policy. Bitran an onschein (997) stuy a similar problem with non-time homogenous eman an emonstrate the effectiveness of restricting the number of prices to a small set an enforcing monotonic polices with respect to the price over time. Zhao an Zheng (2000) stuy an extension of Gallego an van Ryzin (994) with non-homogenous reservation prices. Finally, Feng an iao (2000) fin an exact solution for the multiple-price moel in continuous time when monotonic pricing policies are assume. Our work is relate to this work as we search for allocation policies in a multiple-price moel, where we explicitly incorporate the time-imension. However, our moel iffers from this work in that we assume we can observe the eman in each perio prior to making an allocation ecision. Further, while previous papers have assume reecte customers are lost, we allow for reecte customers to wait, i.e., they are (partially) backlogge with a probability epenent on a price iscount. Outsie of the yiel management context, there has been work on integrating ynamic pricing with prouction/supply policy but mostly for single prouct homogenous customer populations (e.g., Zabel (972), Thowsen (975)). In more recent work, Chan, Simchi-Levi an Swann (200) consiers a multi-perio eterministic 4

6 eman moel where pricing an prouction ecisions must be mae for each perio with some capacity constraint. Similarly, Feergruen an Heching (999) in a stochastic eman setting focuses on showing optimality of a policy where a base-stock is orere up-to an a list price is charge. The remainer of the paper is as follows. In Section 2 we formally introuce the moel. We present a solution to the problem in Section 3. In Section 4 we consier the continuous-time version of the problem. In Section 5 we iscuss the solution an provie some numerical example. We raw our final conclusions in Section PROBLE FORULATION Base on the problem escription, we consier a moel in which there are K customer classes an stages in each perio. Let subscript i I = {,, K} enote the customer class an subscript {,, } represent the stage. Let the per unit revenue from class i be p i an, without loss of generality, assume p p K. Let i be the eman from customer class i in stage an let {,, = K} be the eman vector in stage. We assume that each customer orers exactly one unit. We assume that the eman istribution in each stage is known an inepenent of previous stages. Let enote the supply at the start of stage an let Y be the inventory allocate in stage to eman. Let z be the iscount offere to a class- i customer in stage if they are not i allocate supply promptly an let z {,, z zk} =. Thus a customer, initially enie supply, will pay p i z i if he commits to wait for elivery. Let the probability that a customer will wait be γ i an we assume for simplicity that it is efine by the linear function γ = α + β z i i i i That is, those customers not receiving supply choose to wait base on inepenent αi Bernouli ranom variables. We restrict 0 zi so that 0 γ i. We also assume βi that α i an β i are non-ecreasing in i, i.e. where 0 αk α an 0 βk β. Thus customers with lower prices are more likely to wait. If a customer chooses to wait, the firm may still allocate supply to that customer in a subsequent stage or may not allocate supply until the en of the perio. 5

7 We aopt the following costs. The per unit cost to the seller is c p. If the number of units allocate in stage is larger than a given elivery capacity, g, the firm incurs a per extra unit elivere marginal cost, c, which we refer to as a congestion cost. Such cost results from unplanne overloa of resources an may result from utilizing alternate facilities or aitional resources often at a premium rate. Further, we assume that an aitional cost c w is incurre for each customer that waits for supply until the next perio (e.g., in a istribution center, fulfillment oes not occur until the next ay or after the next elivery from the supplier.) This cost is analogous to a backorer cost. Let c l be the cost of lost eman for those customers that choose not to wait. We let h be the holing cost for units at the en of the perio. We assume pi + cl cp cw 0 an c p h so that prouction an inventory holing are profitable. We also assume that c w + h c so that any units that coul be elivere at the last stage (en of the perio) are not elaye elivery for next perio ue to unrealistically high congestion costs. Let i be eman from class i waiting from a previous stage in stage with {,, = K}. Let = {, }. (Throughout the paper, we use the notational convention of symbols with an arrow, i.e., i, refer to eman from class i waiting from previous stages, while symbols without an arrow refer to new eman from class i arriving in stage. Symbols with a bar, i.e., i, refer to all emans, i.e., both new an waiting eman.) K Let D = ( i + i) i= K i= i, the total new an waiting eman in stage. Let D = be the total new eman in stage. K Let D = be the total waiting eman in stage. i= K i Let Di = k= i k be the new eman from classes i an higher in stage. K Let D = be the waiting eman from classes i an higher in stage. i k= i k Ω be the set of all permutations of (,, D ) Let an let ω Ωbe some orering of the eman, both new an waiting, in a stage. Let D ω li be the position in the orering ω that the l th customer from class i arriving in stage is to be serve in stage an let D ω li be the position in ω that the l th eman from class i waiting from a ω ω ω = D, D for i =,, K an previous stage is to be serve in stage. Then { li li} l =,, i (or i ) efines a priority orer of service for the eman D. 6

8 Let li ( Y, z,, ) new class i customers an let πli ( Y, z, ω, ) π ω be the margin obtaine from the l th unit of eman from be the margin obtaine from the l th unit of eman from waiting class i customers for an allocation, iscount an priority orer given the inventory an eman. Then, ω pi Y Dli πli ( Y, z, ω, ) = () ω γi ( pi zi cp cw ) ( γi ) cl Y < Dli ω cp + cw Y Dli an πli ( Y, z, ω, ) =. (2) ω 0 Y < Dli Π, be the optimal expecte profit from stage onwar given inventory Let ( ) an eman vector. The expecte profit for stage onwar for a given Y, an ω, is π Y, z, ω, ( ) K i i = πli ( Y, z, ω, ) + πli ( Y, z, ω, ) (3) i= l= l= + c( Y g ) + E Π (, ) Observe a congestion cost is incurre for the units allocate over the capacity, g i. + x = max 0, x. Also let the inicator function, x = if x is true, 0 otherwise.) The problem the firm faces at stage is (Note ( ) ( ) (, ) max π ( Y, z, ω, ) Π = Y, z, ω s.t. ω Ω ( ) z (4a) (4b) Y min, D (4c) + Y (4) zi ( αi ) βi, for all i (4e) Y, z 0, for all i (4f) i We efine the en of the perio profit, (, ) ( ) Π = c h to be the p + value from the units hel to the next perio, an note that we have alreay accounte for the expecte revenues from waiting eman. With no initial waiting eman, the profit for the multiple stage problem is max π ( ) = cp + EΠ (, ) (5) 0 7

9 We refer to the problem of etermining the inventory allocation, the customer iscounts an the prioritization of eman for all stages as the Allocation, Discount an Prioritization (ADP) Problem. 3. ADP PROBLE SOLUTION We solve the problem through ynamic programming starting first with the final stage an then solving the problem by inuction for stages,,. 3. Stage Determining the Discount Proposition : Given (, ) where z i For stage, we have the following:, the unique optimal price iscount is ' 0 zi < 0 ' ' zi = zi 0 zi ( αi ) βi (6) ' ( αi ) βi zi > ( αi ) βi p ' i + cl cp cw αi zi = for i =,, K. (7) 2 2β i Proof. From (3), the first orer conitions for the unconstraine problem max π ( Y, z, ω, ) are z π z i = = i l= ( αi + βi zi )( pi zi cp cw ) ( α β z ) lst :..: Y < Dli i lst :..: Y < Dli ( ) = β p + c c c α 2β z = 0 π z li i i i i l z i i l p w i i i c for all i =,, K. which gives z ' i as above. Constraining the solution so that 0 provies the result. γ i 8

10 Let γi = αi + βi zi. Observe that the iscount z i offere oes not epen on, Y, or ω. This allows us to etermine ω an Y inepenently below. Priority Determination (per unit) eman enial penalties, ω an Allocation Determination Y. We efine the L i an L, to be the reuction in profit if a eman from class i or one of the waiting emans, respectively, cannot be satisfie in stage. Following the evelopment in () an (2): L z, ω Y, an ( ) i i ( ) ( γ ) γ ( ) = p γ p z c c + c i i i i p w i l = p + c p + c z c c i l i i l i p w L z, ω Y, = c + c. ( ) i p w We make the following claim: Claim : A priority sequence in stage is optimal for all allocations Y if an only if the eman enial penalties are non-increasing with the orer of the priority sequence. ' Proof. Assume ω is an optimal priority sequence. Consier a pair of emans with D = D +, that is, the l th customer of class i is followe by the l 'th customer of l '' i li class i ' in ' ω. (For purposes of presentation, consier the set of waiting customers ust Li' > Li an Y = D li. Then the profit of the sequence can another class.) Suppose that be improve by interchanging the two in the eman sequence which is a contraiction. Similarly, suppose L i' L i if Dli '' > Dli for all ii, ', ll., ' Then choosing any set of emans an rearranging them cannot ecrease the cost for any Y, therefore, ω with a enie eman penalty non-increasing is optimal. (8) (9) Base on the claim, we have the following: Proposition 2: A priority sequence with customers in serve in class orer, from K to prior to serving any waiting eman,, is optimal. Proof. Observe that L i' L if i' > i since i 9

11 since = ( + γ ( + )) ( pi + cl γ i ( pi + cl zi cp cw) ) ( pi + c ( )) ' l γ i ' pi + c ' l zi ' cp cw ( pi + cl γ i ( pi + cl zi cp cw) ) z i maximizes i ( z)( pi cl z cp cw ) L L p c p c z c c i ' i i' l i ' i' l i ' p w 0 γ = α + β z (c.p. Proposition ). i i i i " zi zi min zi', αi βi Letting ( ) γ + conitional on " = ( ) implies γi min( αi βi zi',) Since αi αi' an βi βi' an p + c c c z from Proposition, i' l p w i' γ, i ' " " γ (( ) ( )) " ( γi γi )( pi ' + cl zi' cp cw ) L L p p + p z p z + i ' i i' i i i i i' i ' 0 +. γ α + β z = γ, an " i i ' i ' i ' i ' Also observe L L L L = p + c γ p + c z c c c + c Therefore ( ) ( ) i l l p w p w 0 ω serves new customers in class orer prior to serving any waiting customers.. Proposition 3: Allocating the inventory to all of the eman at the en of perio, i.e. Y = min, D, is optimal. ( ) Proof. The result follows irectly from the assumption cw + h c. Concavity of E (, Π ) Proposition 4: ) E [ ] an E [ Π ] c h. p 2) E Π D is ecreasing in cp + cw E Π D 0. Π is ecreasing in, increasing in D, continuous D, increasing in, continuous an Proof. The profit rate function is 0

12 (, ) Π π ( Y, z, ω, ) K i i πli π li = + i= l= l= + + c ( min (, D ) g ) ( D ) ( cp h) + + LK ( zi, ω, ) ( c ) g > DK = L ( z, ω, ) ( c ) g D 2 < D cp + cw ( c g ) D < D + D cp h D + D < Π an Π is ecreasing in from (8), (9) an the So that cp h assumption cw + h c. Uner the assumption that the eman is continuous, taking expectations provies the result in ). A similar approach provies the results in 2) as E (, ) D Π = ( h+ cw) P( > D ) cp( > D > g ), an by observing that the absolute value of the partial erivative of the secon term is ominate by that of the first term Stage We now consier the problem face at stage. We show by inuction how the iscounts to offer, z i, the priority to assign to customers in allocating inventory ω, an the amount of inventory to allocate Y, are foun. We show that, as in the stage solution, the iscounts to offer can be foun by comparing the incremental revenue receive after price iscounting, though now an algorithm is require to fin a feasible optimal solution. We then show that the optimal priority sequence is inepenent of these iscounts an orers new customers first (in class orer) followe by waiting eman as in the stage solution. Finally we show how to etermine the amount of inventory to allocate in stage.

13 We assume for the inuction that for ' >, EΠ ' ( ', ' ) is concave in ' an D ', that E Π ' ' is increasing in D ', continuous an E Π ' ' cp h an that E Π ' D ' ' is increasing in ', continuous an cp + cw E Π ' D ' 0 '. Note that this implies that holing the remaining units in inventory is profitable an, without loss of generality, we have = Y 0 in the rest of our paper. + Determining the Discount z i Let p ' i + cl cp cw αi Π zi = + E + 2 2β i 2 ( Y, ) (0), D Proposition 5: In stage, ' z i maximizes ( Y, z,, ) π ω. Proof. π z i ( Y, z, ω, ) i ( Y, z, ω, ) E ( Y, + + ) = πli + Π + zi l= z i D+ Π + = ( βi ( pi + cl zi cp cw ) γi ) ci + E ( Y, + ) + zi D+ where c i is the number of customers of class i enie service in stage. Letting γ i = αi + βizi an observing D + zi = βici implies αi 2βi zi + βi ( pi + cl zi cp cw ) π = ci Π () + zi + βie ( Y, + ) + D + Noting π zi is ecreasing in z i since D + increasing in z i an E Π + D + + ecreasing in D + from the inuction assumption, setting π z = 0 an solving provies the result. i By comparing (7) with (0) we observe that the optimal iscount maximizes the current profit plus a term that reflects the change in future profits resulting from a change in the total number of waiting customers. 2

14 , Noting that ω an ' z i solves the unconstraine problem, the constraine problem, given Y,, is: π ( Y z ω ) s.t. z ( α ) max,,, We have the following proposition: 0 β :: i,, (2) i i i Proposition 6: Suppose in an optimal solution, that αx α y α x α y 0 < zx < an 0 < zy <, then zxy, zx zy = px py. βx β y 2 β x β y Proof. Consier the Kuhn-Tucker conitions for Problem (2), if z, to (2) for x, y {,, K} α ' y π π 0 zy, = = 0. Then π π = 0 β y zx zy βxcx zx βycy zy α x α y 2zx + px 2zy + py = 0 which gives the result. β x β y α ' x 0 zx an β x implies, by () The proposition implies that if we sort the classes by p i α i β i an increase z i from 0 for each class in this orer, while maintaining a separation of zi an ensuring that no z i excees its upper boun, we will eventually fin an optimal solution to the constraine problem. We formally efine this through the following algorithm: Algorithm : Discount z i Determination Algorithm Initialization Let ri = pi αi βi an bi = ( αi) βi. (Throughout the algorithm we suppress the stage subscript for ease of presentation) Orer the classes by r i. (For purposes of clarity, we assume without loss of generality, that r r K.) Set zi = 0 : i. Set x = (x is the last class entering the active set, i.e. the set of classes for which z i is not at its upper or lower bouns). Let A= { }, be the set of customer classes that are active (incluing a class entering the active set), Set ( ) for x K z =. Step. Repeat { } zx rx r x + 2 < an K A: = A x+, x : = x + while 0 { yby zy b} z x =. Set b min y ℵ ( b y z y ) =. Let R= =, be the set of active classes that woul next achieve their upper bouns. Set z ( b z ) = min, x. 3

15 Step 2. Test: Is there z ' ( 0, z) ' ' such that by letting zy = zy + z': : y A, z y solves (0) for one class in A. If so, let z : = z + z': : y A an stop. (The solution is optimal at the current levels of z y.) Otherwise, go to Step 3. Step 3. z : = z b. z : = z + z: : y A. A: = A\ R if z : = b. x x y y y Step 4. If A = an x = K, stop. (All z x are at their upper bouns.) Step 5. If A an z x > 0, go to Step. Step 6. A: = A { x+ }. x : = x +. Go to step. In the algorithm the value of z i is successively increase from 0 until either the value which is active (i.e., in set A ) solves the first orer conition or reaches its upper boun. Since a spacing of z is maintaine between active classes, Proposition 5 hols for x active classes. In each iteration either a class enters or at least one class exits the active set, so that there are at most 2K iterations. In each iteration the number of tests (in Step 2) is O(log z) since π zi ecreases in z i from Proposition 5. This results in the following: Proposition 7: In stage, given Y,, y ω an z is unique an given by the Discount Determination Algorithm., the optimal price iscounts vector Determining ω Following the evelopment for the stage case, we efine the (per unit) eman enial penalties base on (), (2) an (3) L Y, z, ω, an ( ) i i = p + c γ p + c z c c E Π Y + ( ) γ (, + ) i l i i l i p w i + D + (3) Π + L( Y, zi, ω, ) = cp + cw E ( Y, ). + (4) + D + Claim 2: L i an L i are continuous, increasing in D an ecreasing in. Proof. We observe the results from (3), (4) an the inuction assumptions E 0 Π + D + < + an E Π + D + + increasing in. 4

16 Claim 3: A priority sequence ω is optimal for all allocations Y if an only if the eman enial penalty in stage is non-increasing with the orer of the priority sequence. Proof. Similar to Claim. Proposition 8: A priority sequence with customers in serve in class orer, from K to, prior to serving a waiting eman,, is optimal. Proof. Let + enote the (ranom) total eman vector in stage + uner Observe that Li' Li if i' > i since (suppressing the subscript for conciseness) L L i' i z = z. Π + pi' γ i' pi' + cl zi' cp cw + E (, Y + ) + D + Π + pi γ i pi + cl zi cp cw+ E ( Y, + ) + D + γ = α + β z for any z i such that 0 zi ( αi) βi since z ' i maximizes Π + z p + c z c c + E Y, + an ' z i equals either z i or 0 or + D + Π + α β E, Y +. + D + " Letting z, i min zi' ( αi) β " i γ min α + β z, α + β z = γ since for i i i i γ ( ) ( ) i i l p w ( ) i i from the monotonicity of ( ) ( ) implies ( ) αi αi' an βi βi'. Therefore L L γ p p + γ z z " " " ( )( ) ( ) i' i i i' i i i' i i i i i' i' i' i' i' + Π D+ 0 ( " ) + γ ( i γi pi' cl zi' cp cw E Y, + ) Π by Proposition 7 an + D + pi' + cl cp cw + E Π D from the inuction assumption E 0 Π + D + + an the assumption pi' + cl cp cw 0. since + z ( i' pi' + cl cp cw + E Y, + ) + 5

17 Determining Y Let Y i be the number of units to allocate in stage if when the last customer to receive a unit is from class i. Similarly, let Y be the number of units allocate in stage when the last customer to receive a unit is among the waiting customers. We have the following two claims regaring these values: Claim 4: Y i is unique an solves π ( Yi, z, ω, ) Yi Π + = pi + cl γ i pi + cl zi cp cw + E (, Y + ) + D + Π + ( c Y ) (, ) g E Y Π + = Li ( c Y ) (, ) g E Y = 0 an Y is unique an solves π Π + ( Y, z, ω, ) = L ( c Y ) (, ) 0 g E Y + = + Y + π Proof. ( Yi, z, ω, ) Y i is the marginal profit when a unit in is shifte from stage + to stage an is use to satisfy a eman of class i. Consiere as two operational steps, ) increasing an unit on an allocating it to a new eman from Li c Y g. 2) reucing a unit from an class i at stage, the marginal profit is ( ) +, the marginal profit is E Π By the inuction assumption, both E Π an E Π + D + + are monotonic an so the solution exists an by continuity of the eman istribution, unique. A similar proof hols for the case that an unit in is shifte from stage + to stage an use to serve a waiting eman in Y. Claim 5: Y i is non-ecreasing in i an Y Y. 6

18 Π Π Proof. = Li+, Li 0 from the proof of Proposition 8 so that Π Yi is Yi+, Yi non-ecreasing in i. Therefore the solution of Π Y = 0 is also non-ecreasing in i. i The actual allocation in stage follows the following inventory allocation rule: Y equals Y i if the equals Y if the Y th i customer is of class i, i.e., Dli Yi an Y customer is among the waiting eman, i.e. Dli = Yi for some i an th = for some l {,, i} l. Finally, if D < Y, i.e., the total eman is less than the optimal amount to allocate for all the waiting eman, then Y equals D. In all cases, Y must be less than or equal to. Since the optimal priority sequence is to serve the new customers in class orer from K to, prior to serving any of the waiting customers, we observe that D is ecreasing in i an so in light of Claim 5 we have the following proposition: li Proposition 9: The optimal number of units allocate in stage is unique an is given by the inventory allocation rule. Proof. Suppose that Dli Yi l,, i an Y i. Then all of the new eman from classes i+,, K are allocate inventory an no eman from classes,, i or waiting eman is allocate inventory in accorance with the priority sequence. By Claim 5 Y i' Y i an by Proposition 8, D l '' i < D li for i' > i an l' {,, i' }. Similarly, for classes i' < i an all waiting eman, we know Yi' Yi an Dli '' > Dli. Therefore Dli '' Yi'. Therefore Y = Yi. A similar argument follows if Dli = Y an Y so Y = Y. Finally if, D < Y an D, then Dli D< Y Yi for all i an D D< Y for all i so that letting Y = D is feasible an optimal, since holing the li = for { } remaine units is profitable from the inuction assumption. In any of these cases, if < Y then from (4c) we know Y =. hol. Having establishe z i, ω an Y, we nee to show that the inuction assumptions 7

19 Proposition 0: ). E Π is ecreasing in, increasing in an E Π cp h. 2). E Π D is ecreasing in cp + cw E Π D 0. i= D, increasing in ( ) i D i+, Y < Di D Y < D g Y D D, continuous, continuous an Proof. If < D, the incremental inventory may be allocate at stage an π ( Y +, z, ω +, ) π ( Y, z, ω, ) lim 0 pi + cl zi cp c w K i = pi + cl γ i Π ω + Y = Dli i= l= + E ( Y, + ) + D + K i Π + + cp + cw E ( Y, + ) ω + li D + Y = D i= l= ( c g ) Y D K = L + L c If the incremental inventory is allocate at stage +, then (, ) E π = Π Since we always allocate the units to the stage with higher incremental profit, then from the inuction assumption, it must be that π K Π + = max Li L ( c ), E D i, Y Di D Y D g Y D + + < < + i= + Π + E cp h Since L i, L an E Π are ecreasing in, increasing in D, an continuous from Claim 2 an the inuction assumption, by taking expectations the result in ) hols. A similar approach provies the results in 2) by Π Π + Π + (, ) = maxcp + cw E ( c ), E + g D + D + D+ 8

20 Let be the optimal initial inventory hel. We have: Claim 6: is unique an solves π ( ) cp E ( ) = + Π, = 0 Proof. The result hols from (5) an Proposition 0 2). We summarize the results in this section as follows: Theorem. For the Allocation, Discount an Prioritization (ADP) Problem, the optimal price iscounts, the optimal priority sequence, the optimal unit allocation policy an the optimal base stock level are unique The Expecte Deman Heuristic The computational time for the optimal ADP solution increases exponentially with the number of the stages an the customer classes. In this section, we introuce the expecte eman heuristic (EDH) which assumes eman equals its expectation in each stage when we etermine the price iscount an allocation rule. EDH First, fin the optimal price iscount at stage, i.e. z, using (6). Let e e Y (, ) an z (, ) enote the solution to the ADP problem at stage, assuming the arriving new eman equals its expectation, i.e., given ( ) = E( ) = ( E(, ),, E( K, ) ). We can etermine ( e e, ) each stage given (, ) an assuming the arriving new eman E( ) e e e e e e e can fin the initial inventory level uner the policy ( Y, z) ( Y,, Y, z,, z ) Y z by inuction for =. We. Since we only calculate the price iscounts an the allocation policy once at each stage, the computational time is reuce significantly. oreover, since Y an z are constant matrices with two imensions (, ) store the values of ( e e Y, z )., the system can make real-time ecisions if we We can show that the EDH is optimal when all eman is lost, i.e. αi = βi = 0 so that D = 0. In this case, the profit function at stage is epenent only on the inventory level from the previous perio an there is no nee to offer a iscount ( z i = 0 ). Further, the eman enial penalty, L i, is inepenent of the eman in the previous perios. e e 9

21 Therefore, consiering the following perio, there exists a unique pair of inventory levels, enote as ( 2 2 i, +, i, + ), where i, + solves E Π + + = Li + an i, + solves E Π + + = Li c +. Therefore, we can use a simplifie allocation rule, calle the critical inventory level rule, where class i in stage receives the last unit allocate in stage if an only if the on-han inventory falls to i, + if Y g or if Y > g. 2 i, + Proposition : For the lost eman case, the EDH approach for calculating price iscounts an the critical inventory level rule are optimal. Proof. Since the price iscount is zero an there exists no waiting eman, the prioritize 2 orering is naturally from class K to. Noting, then similar to the proof of i, + i, + Proposition 9, the optimal number of units allocate in stage is unique an is given by the critical inventory level rule. Since the price iscount an the allocation rule are ientical for any observe eman, the solution of EDH is also optimal to the ADP problem. For the general case, we can emonstrate the effectiveness of the EDH for a two-,, the price iscount error is stage problem. Given ( ) z = z z z z e ' ' e i i i i h+ c (5) w e e = P max ( D 2, D 2) Y D 2 > min ( D 2, D 2) 2 ' e e where z an D 2 is the global optimal price iscount an the waiting eman of EDH, i respectively. The error on the profit rate function is Π π π e e = Yi, z, ω, Yi, z, ω, Yi Yi Yi ( ) ( ) π π = Yi Yi e e max D2, D2 E( ) e = ( ri )( h+ cw) P Yi D 2 e e > min D2, D2 E( ) e e e e ( Yi, z, ω, ) ( Yi, z, ω, E( ) ) Because the likelihoo of eman waiting is base on the iscounts which are calculate optimally in the EDH given the expecte eman, the probabilities (6) 20

22 ( 2 2) an P ( D2 Y e D2) P D Y D are close to each other for most practical cases. Thus, the probability terms in (5) an (6) are very small. This implies that the heuristic is quite accurate, even though the exact error boun cannot be clearly establishe. To illustrate the accuracy of the EDH heuristic, we present computational results in Table. We experiment with the following ata set: c p =, h = 0.6, K = 2, α = ( 0,0) an p ( p,2) =,0., =, 2. All emans i, i =, 2 an =, 2, are inepenent with each other an follow the same normal istribution with µ = 300 an =, β ( β ) σ = 90. We present results by varying the parameters p an β. As it can be seen both the EDH expecte profit an the initial inventory ecision are very close (less than 2% eviation for the profit an less than % eviation for the initial inventory) to those of the optimal ADP solution. (The same results hol over larger numerical sets we experimente with, but not reporte here for brevity of presentation reasons). Parameters Expecte Profit Base Stock β ADP EDH Error ADP EDH Error p % % % % % % % % % % Table : Accuracy of expecte profit an base stock level for EDH 4. An Approximating Heuristic for the Continuous Time Deman Case In most cases, emans arrive accoring to a continuous time stochastic process an firms are require to respon to emans immeiately, either accepting the orer or offering a iscount to encourage the customer to wait for elivery in the next perio (or, of course, reecting the eman). We can approximate such a case by iviing the perio into a large number of stages,, of, say, equal uration. However, in making this approximation, we nee to assume that the congestion cost, c = 0, because the capacity in each stage, g i, becomes increasingly small as increases. Therefore, with no 2

23 congestion cost, it is easy to see that, in an informal sense, there is less information available when a ecision is mae regaring iscounts. Because of this the profit for the firm woul ecrease as increases. Because of the computational ifficulty in solving the continuous time problem optimally for a large number of stages, we consier a heuristic that etermines whether a eman shoul be accepte base on the current inventory, waiting eman an the assumption that all future emans will occur in a single stage. That is, consier a single arrival from class i at t. Let () t an () t enote the inventory level an the waiting eman at time t. Let k ( t, T ] enote the ranom variable of eman from class k customers in the time interval ( tt, ] for all classes. We consier the following two-stage heuristic (TSH): Consier a two-stage ADP problem where the first stage has ust been complete with i = an i' = 0 for i' i, = () t, = () t, i2 = i( t, T]. That is, there is only the arrival from class i to consier for allocating the current inventory an the eman for the secon stage is assume to be istribute as the total of all future emans. Because the congestion cost is 0, the waiting eman is only allocate inventory at time h T. Then we can calculate a price iscount in TSH, enote as zi () t, is accoring to (6) an (0) an Proposition 7. pi + cl cp cw αi () t h 2 2β () () i () i t α t zi t = max min,,0. K βi () t + ( h+ cw) P( ( t) > ( t) + (, ] k k t T = ) 2 We then allocate a unit of () t to the class- i customer if an only if () t i t, () t, where t, () t is the unique solution of i h ( α () β () ()) p + c t + t z t = i l i i i h i + l i () p w K ( h c ) P ( t) ( t, T] p c z t c c + + > + ( k = ) w i k h h K pi + cl ( αi( T) + βi( T) zi ( T) )( pi + cl zi ( T) cp cw) K i= P ( tt, ] > ( tt, ] + ( cp + cw) P k ( tt] + () t i > k ( tt] + ( cp h) P i < k t as shown by Claim 4 where K ( k= i k i k= i+ k ) K K (,, k= k= ) K ( (, T] + () t ), k = 22

24 ( ) ( T) ( ) ( T) pi + cl cp c h w αi T αi T zi ( T) = max min,,0. 2 2βi β i h It can be shown from the formulation of zi () t that the heuristic has the properties that the offere iscount increases with greater inventory an ecreases with greater waiting eman. Further, as the number of waiting emans increase, the cut-off value t, increases. The base stock level for the TSH is given by solving the single stage problem assuming that eman for each class in the stage is istribute accoring to the eman T. for the entire perio, i.e., as in ( ] i, 0, The computational time of TSH is almost real-time, since the consiere two-stage problem is simplifie to a moifie single stage problem. Because the values of z () t () t, () t an t, () t can be store for boune eman functions, the h i i heuristic can in practice be applie in real-time. Intuitively, the TSH shoul be very close to the optimal solution, since it can aust the allocation an elivery ecision solution accoring to the observe eman information. Observe that the optimal expecte profit for a single stage problem in which a single ecision is mae after observing all of the eman is an upper boun of the expecte profit for the TSH since the inventory allocation is globally optimal. To illustrate the effectiveness of the TSH heuristic, we present computational results in Table 2. We experimente with the following ata set: c p = 0, h = 8, K = 2, α = ( 0,0) an p= ( p,20), β = ( β, 0.0). The class eman () t an () t, t ( 0,00], follow a pair of inepenent Poisson istributions with arrival rates λ = 0.9 an λ 2 = 0.. We again varie the parameters p an β. From Table 2, it can be seen that TSH was within less than 0.5% from the upper boun to the optimal ADP expecte profit (upper bouning approach escribe in previous page with the use of the one stage problem). Further, the TSH solution resulte in total fill rates per class (i.e, total eman per class eventually met) that were within 2% of those provie by the optimal ADP solution, with the TSH solution provie higher fill rates for class- an lower for class-2 than it is in the profit optimizing solution. (The same nature an magnitue of performance hols over larger numerical sets we experimente with, but not reporte here for brevity of presentation reasons). i 2 Parameters Expecte Profit Total Fill Rate Per Class β ADP TSH ADP / i = 2 TSH / i = 2 ADP / i = TSH / i = p 23

25 % 99.07% 00% 00% % 98.47% 00% 00% % 98.% 99.52% 99.66% % 98.9% 96.99% 97.24% % 98.08% 96.98% 97.9% Table 2: Accuracy of expecte profit an total fill rate per class performance. 5. A Comparison of the ADP Solution to a FCFS Policy In this section we compare the optimal solution of the ADP to a First-Come/First- Serve (FCFS) policy. This iscussion illustrates the role of ynamic inventory rationing an price iscounting in the optimal solution. 5.. First-Come-First-Serve (FCFS) Policy The First-Come-First-Serve policy is a simplifie ADP policy without inventory rationing consieration. We use the superscript f to ientify the variables an functions uner the FCFS policy throughout the paper. The profit for the two-stage problem is f f f π ( z, ) f ( D ) min, = D ( min (, ) ) K f p i i c i D g = f + ( D ) K f f f + i i ( pi zi cp cw ) cl ( i ) D γ γ i= f f + + E { Π ( ), D } f f f f where Π (, ) ( cp h)( D ) Observe that uner this policy there exist no inventory rationing consierations, an we offer price iscounts only if there is no inventory left. The problem the firm faces at stage is f f f f f Π, max π ( z, ) (8) ( ) f α i 0 zi βi The profit for the multiple stage FCFS problem is f f f f f max Π ( ) c ( ), f p + E Π (9) 0 + (7) 24

26 Proposition 2. The unique optimal price iscount at stage of the FCFS problem is z max min,,0 p i + cl c f p cw αi αi i = 2 2βi βi (20) Proof. Similar to Proposition. f f f Let γi = αi + βizi. Observe that the iscount z i offere oes not epen on,. This allows us to etermine base stock inepenently below. Proposition 3. If L E f i i f c f D k k> D k k D ( D k k) g = = = ecreases in, the optimal f FCFS base stock is unique an solves f f Π ( ) f = cp h P f ( D = ) f P( D k > D k k k) = = + f K L i i = E c f i f D ( k k> D = k k D D k k) g = = = = 0 f f f f where = γ ( ) ( γ ) L p p z c c c i i i i i p w i l Proof. The results follow from consiering the first orer conitions for the problem in (9) The Role of Dynamic Priorities, Inventory Rationing, an Price Discounts We compare the expecte profits, base stocks ( ) an eman waiting rates (γ ) for the FCFS solution to those of the optimal solution to the ADP problem. Because the FCFS solution is a feasible solution to ADP, its expecte profits may be at most as large as those of the optimal solution. We observe that the optimal solution is achieve by both allocating inventory to the most profitable customers through their prioritization an by offering iscounts to retain others. In the FCFS solution, only the latter, i.e. iscounting, is retaine. 25

27 As the following example illustrates, the comparison of base stock levels an eman waiting rates between FCFS an optimal ADP solution is non-conclusive. The optimal ADP solution may either increase or ecrease, vis-à-vis the FCFS solution, the base stock levels an the eman waiting rates. Example : Assume that =, K = 2, an customer class emans are inepenent normal ranom variables, ~ (, ) ( ) 2 ED> i D = an, from Proposition 4 an 3, (, ) Π (, ) 2 N µ σ. We have D ~ N 2 ( 2 µ, 2σ ) f Π E E K K = P D> L E LP D > D f i ( ) ( ) i D> i i i i= D i= ( ) ( > ) ( ) P D> P D> = L P( D > 2) + L2 P( 2 ) 2 2 P D = ( L2 L) P( 2 ) 2 f Thus, f 2 f P D > P. if an only if ( ) ( ) For the case with f = +,, if α = α 2 an β = β 2, we have f f γ 2 = γ2 γ = γ from Proposition an 2. The eman waiting rate uner FCFS may be higher in certain cases than uner the ADP policy. We now look at the ADP an FCFS solutions without price iscounts, i.e. given z = 0, or equivalently with β = 0. We have irectly from Theorem : i i Proposition 4: For the multiple-stage ADP problem without price iscount, the optimal priority sequence is to serve the customers in in class orer, from K to, prior to serving the waiting eman,. The optimal allocation policy is unique. ( ) α ( ) ( α ) i D pi i pi cp cw i c l Proposition 5. If E f ' D k k> D = k= k c f ' ( D ) gi ecreases on, the optimal base stock level for the FCFS problem without price iscount is unique. 26

28 We compare the solutions with an without price iscount for the ADP an FCFS solution to illustrate the role of price iscounts. Obviously, the ynamic price iscounts will increase the expecte profit for either the optimal ADP or FCFS solution, as they wouln t be use otherwise. Proposition 6. Use of Price iscounting reuces the profit maximizing base stock level for the FCFS policy. Proof. Given an inventory allocation, the value of E Π with f f zi = zi is f no larger than with z = 0, since the eman enial penalties are minimize when z f i E f i i = z. Thus, price iscounting reuces the optimal base stock level, which satisfies Π = 0. The result in Proposition 6 oes not apply to the general ADP problem Numerical Insights Example 2: Assume that c p =, h = 0.6, 2 ( 0, 0.) β = for =, 2. All customer class emans i,, 2 K =, p = (.2,2), ( 0,0) α =, i = an =, 2, are inepenent with each other an follow the same normal istribution with µ = 300 an σ = 90. The congestion cost in the first stage is c = 0.6 with g = 0 an is zero in the secon stage, i.e. c 2 = g 2 = 0. We also assume cw = cl = 0.0. (Note that non-zero c l are effectively capture as part of the price parameter variations, an the price iscount has a similar effect as c w.) Item name ADP ADP-Z=0 FCFS FCFS-Z=0 Expecte Profit Base stock Discount Cost Total Fill Rate Average 99.98% 65.80% 97.08% 94.46% Class % 99.46% 94.6% 94.46% Class- 00% 32.6% 00% 94.46% 27

29 Prompt Fill Rate Average 7.59% 65.80% 93.94% 94.46% Class % 93.94% 94.46% Class % 32.6% 93.94% 94.46% Table 3: Results of Example 2 The ADP solution not only has the highest profit, but also the highest total fill rates. Inventory rationing increases the profit by 83.26%, reuces the base stock level by 2.4%, an increases the total fill rate of class-2 eman by 6.2%. The average total fill rate is also increase, even though the base stock is reuce. The effect of price iscounts for the case with inventory rationing is much more significant than without inventory rationing. For the case with inventory rationing, price iscounting increases the expecte profit by 4.36%, increases the average total fill rate by 5.95% an the class- total fill rate by 20.95%. Also, note that price iscounting increases the base stock for the case with inventory rationing, but has the reverse effect for FCFS policies. While price iscounting improves class-2 total fill rate for inventory rationing policies, it ecreases the corresponing fill rate for FCFS ones. Figure : Expecte Profit (y vs. x) Deman Variance ADP FCFS ADP-Z=0 FCFS-Z=0 28

30 Figure 2: Base Stock (y vs. x) Deman Variance ADP FCFS ADP-Z=0 FCFS-Z=0 Figure an 2 report the effects of eman variance on expecte profits an base stock levels for our various policies on the ata set of Example 2. All the expecte profits might ecrease with the eman variance as is intuitive. For the optimal solution to the ADP, the profits are quite stable (profit reuction in most examples less than 0.%) since most of the class 2 customers are serve promptly an the cost of enying immeiate eman satisfaction eman to the class- customers is small. The base stock of the FCFS solution ecreases more rastically with an increase in variance, as woul be expecte. Similar to a newsvenor-like formulation an for a base stock level less than the mean eman (as is the case in this example), an increase in the eman variance reuces the base stock level. We observe that the base stock for the optimal solution oes not necessarily ecrease with an increase in variance. Because of customer class prioritization, an increase in the variance increases the marginal contribution from the class-2 customers (i.e. the prouct of the probability of an aitional unit being allocates to class 2 customer times the profit margin of this customer), but only by a little as it is very close to zero at the optimal base stock level, but ecreases the marginal contribution from the class- customers (as oppose to both customer class marginal contributions ecreasing in the FCFS solution). The combine effect results in at first increasing an later ecreasing with increasing variance of the base stock level. 29

31 Figure 3: Expecte Profit (y vs. x) Class Deman Ratio at Each Stage : 3;3: 9 8: 4; 4: 8 7: 5; 5: 7 6:6;6:6 5:7;7:5 4:8;8:4 3:9;9: 3 ADP FCFS ADP-Z=0 FCFS-Z=0 Figure 4: Base Stock (y vs. x) Class Deman Ratio at Each Stage : 3;3: 9 8: 4; 4: 8 7: 5; 5: 7 6:6;6:6 5:7;7:5 4:8;8:4 3:9;9: 3 ADP FCFS ADP-Z=0 FCFS-Z=0 In Figures 3 an 4 we change the ratio of the eman of the class-2 to class- customers in the first an secon stages an present the expecte profits an the base E E, for =, 2, from = to = 2, an stocks. The ratios are reporte ( 2 ) ( ) show an increasing ratio of class-2 eman in the secon stage reaing the figures from left to right. (Note in these examples we let the mean total mean eman in each stage be 200 an maintain a stanar eviation of 90 for the eman from each class in each stage.) For the optimal solution to the ADP, the expecte profit increases with the increasing eman 2 in the secon stage because fewer units are allocate in the first 30