US20080040202A1 - Generating an Optimized Price Schedule for a Product - Google Patents

Abstract

Generating a price schedule involves generating a graph having paths that include states with values. The graph is generated by determining the values of a successor state from the values of a predecessor state. An optimal path is selected, and a price schedule is determined from the optimal path. Computing an elasticity curve involves having a demand model, values for demand model, and filter sets that restrict the values. Elasticity curves are determined by filtering the values using filter sets, and calculating the elasticity curve using the demand model. A best-fitting elasticity curve is selected. Adjusting a demand forecast value includes estimating an inventory and a demand at a number of locations. An expected number of unrealized sales at each location is calculated. An sales forecast value is determined according to the expected number.

Description

CLAIM OF PRIORITY
• This application is a divisional of U.S. patent application Ser. No. 09/896,388 filed on 28 Jun. 2001 entitled “GENERATING AN OPTIMIZED PRICE SCHEDULE FOR A PRODUCT”, and which claims the benefit under 35 U.S.C. § 119(e) of U.S. Provisional Application Ser. No. 60/238,676, filed Oct. 6, 2000. U.S. patent application Ser. No. 09/896,388 and U.S. Provisional Application Ser. No. 60/238,676 are commonly assigned to the assignee of the present application. The disclosure of related U.S. patent application Ser. No. 09/896,388 and U.S. Provisional Application Ser. No. 60/238,676 are hereby incorporated by reference into the present disclosure as if fully set forth herein.
BACKGROUND
• 1. Technical Field of the Invention
• This invention relates in general to commercial environments, and more particularly to generating an optimized price schedule for a product.
• 2. Background of the Invention
• An important decision for a business to make is determining the prices of products, goods, or services offered by the business. Pricing decisions are important to a business's success or failure, since these decisions may have a critical impact on customer demand, profitability, and business operations. Informed pricing decisions, however, may be difficult to make, since these decisions may need to reflect a large number of business objectives and operating constraints, and may need to be made for a large number of items across a large number of sales locations. Moreover, pricing decisions may need to be frequently updated in order to adjust for rapid changes in business conditions, such as changes in inventory, demand, or a competitor's prices. Consequently, businesses may benefit greatly from price optimization, which may allow businesses to make effective pricing decisions.
• A business may generate a price schedule over a time horizon, and may optimize the price schedule in order to maximize the benefit of a product to the business. For example, a price schedule may be optimized to maximize expected profits or revenues over a time horizon, subject to operating constraints such as an allowed frequency of price changes, a maximum number of price changes, or an inventory target level. Optimizing price schedules, however, may be computationally difficult. As a result, previous techniques for generating an optimized price schedule have been inadequate.
SUMMARY OF THE INVENTION
• In accordance with the present invention, disadvantages and problems associated with generating price schedules have been substantially reduced or eliminated.
• In accordance with an embodiment of the present invention, a method for generating a price schedule includes generating a transition graph having paths. Each path has states, and each state has a price value, an inventory value, and a state value. The transition graph is generated by repeating the following for a number of stages until a last stage is reached. The price value of a successor state is determined. The inventory value of the successor state is calculated using the price value and the inventory value of a predecessor state. The state value of the successor state is calculated using the price value and the inventory value of the predecessor state. After the last stage has been reached, an optimal path is selected according to the state values of the states, and a price schedule is determined from the optimal path.
• In accordance with another embodiment of the present invention, a method for computing an elasticity curve includes selecting a demand model having a number of variables. A number of values for each variable are received. Filter sets that restrict the values for at least one variable are defined. An elasticity curve is determined for each filter set by filtering the values for at least one variable using the filter set, and calculating the elasticity curve from the filtered values by performing a regression analysis using the demand model as a regression equation. A quality value is measured for each elasticity curve, and an optimal elasticity curve is selected according to the quality values.
• In accordance with another embodiment of the present invention, a method for determining a sales forecast from an estimated demand and inventory includes defining a number of locations. An inventory and a demand are estimated at each location. An expected number of unrealized sales at each location is calculated using a difference between the demand and the inventory at the location. A sales forecast is determined using the expected number of unrealized sales.
• Certain embodiments of the present invention may provide one or more technical advantages over previous price scheduling techniques. Selling goods typically requires dynamic adjustment of a sales price over a time horizon. Given a forecast of product demand over time and a model of price elasticity, the present invention may calculate a price schedule that maximizes a business objective such as cumulative revenue, while satisfying business constraints such as a desired price behavior or an inventory target level.
• The present invention may generate price schedules that satisfy a variety of operating constraints. Price schedules are typically required to meet operating constraints such as a maximum number of price changes within a given time period of a time horizon, a maximum number of price changes over a time horizon, or a maximum price change over a time horizon. Price values may be constrained to meet specific price rules, for example, a price value must have a “9” in the cents position. Price values may be constrained to adhere to certain price reduction levels, such as 25%, 50%, or 75% off a base price. Constraints may be applied to the inventory age as described by the time the products are stored in inventory. The present invention may generate price schedules that satisfy these and other operating constraints.
• The present invention may allow for optimization of a variety of business objectives. Objectives may include, but are not limited to, margin and revenue considerations, opportunity costs of capital, inventory carrying costs, inventory age, costs of implementing price changes, impact on profits of substitute products, or opportunity costs of allocated shelf-space, or any other suitable objective.
• The present invention may be extended to incorporate an uncertainty of a demand forecast or an elasticity model in order to achieve certain statistical guarantees of performance of computed price schedules. The invention may be extended to compute price schedules of portfolios of products while considering price constraints between these products, or price schedules for groups of locations given price constraints between locations in the group.
• The present invention may efficiently generate an optimized price schedule using a “quantized dynamic programming” technique. A transition graph with paths representing possible price schedules may be generated using quantized dynamic programming. An optimal path of the transition graph is selected in order to determine an optimized price schedule. The present invention may improve the efficiency of the optimization by quantizing values associated with the transition graph at various degrees of granularity. Quantizing the state values reduces the size of transition graph, which may improve the processing time and power required to determine an optimal path of the transition graph.
• The present invention may take into account the elasticity of a product in order to improve the accuracy of the optimization. The elasticity of a product describes a change in the demand of a product in response to a change in the price of the product. The present invention may take into account the effect of unrealized sales due to supply limitations, which may improve the accuracy of the optimal path. An unrealized sale is predicted where there is a demand for a product, but no inventory, and may result when inventory is distributed across a group of locations that are being optimized together. Typically, aggregate values only take into account a total demand and a total inventory and do not account for supply limitations.
• The present invention may be used to compute an optimal sequence of product markdowns for liquidating merchandise, which may enable sellers to better plan and more profitably manage inventory, such as at the end of a product lifecycle. Given initial inventory and price values, a price schedule of product markdowns that maximizes business objectives while satisfying business constraints may be calculated. The price schedule may be used to maximize margin capture rates while reducing margin degradation and to provide accurate estimates of both margin capture rates and margin degradation factors. In addition, the present invention may be used to determine optimal start and end dates of a markdown horizon, or to determine which products should be put on a markdown plan. The present invention may be used to guide the decision of how many units of which product should be acquired for subsequent sale.
• Systems and methods incorporating one or more of these or other advantages may be well suited for modem commercial environments such as those associated with marking down the price of a product to achieve a predetermined inventory target. Other technical advantages are readily apparent to those skilled in the art from the following figures, descriptions, and claims.
BRIEF DESCRIPTION OF THE DRAWINGS
• For a more complete understanding of the present invention and its advantages, reference is now made to the following description, taken in conjunction with the accompanying drawings, in which:
• FIG. 1 illustrates an example system for generating an optimized price schedule for a product;
• FIG. 2 illustrates an example transition graph that represents possible valid price schedules for a product;
• FIG. 3 illustrates an example method for generating an optimized price schedule for a product;
• FIG. 4 illustrates an example method for computing elasticity;
• FIG. 5 illustrates an example method for optimizing a transition graph to generate an optimized price schedule for a product;
• FIGS. 6A through 6C illustrate example an elasticity curve; and
• FIG. 7 illustrates an example method for adjusting an aggregate inventory value.
DETAILED DESCRIPTION OF THE INVENTION
• FIG. 1 illustrates anexample system10 that generates an optimized price schedule for a product. A price schedule lists prices for a product at successive stages of time. Acompany20 may usesystem10 to determine an optimized price schedule for a product sold bycompany20. Depending on the nature ofcompany20,company20 may include any combination of locations24, for example, one or more sales locations24.
• Company20 and/or one or more locations24 may include acomputer system26 that may share data storage, communications, or other resources according to particular needs.Computer systems26 may include appropriate input devices, output devices, mass storage media, processors, memory, or other components for receiving, processing, storing, and communicating information according to the operation ofsystem10. As used in this document, the term “computer” is intended to encompass a personal computer, work station, network computer, wireless data port, wireless telephone, personal digital assistant, one or more microprocessors within these or other devices, or any other suitable processing device. Reference herein to a location24 may include reference to one or more associated computing devices and/or persons, as appropriate.
• Aserver30 manages applications that generate an optimized price schedule.Server30 includes anoptimizer34 and anelasticity module36. Optimizer includes atransition graph generator32 and aforecast correction module35.Transition graph generator32 may be used to generate a transition graph that represents possible price schedules that satisfy predetermined business constraints.Optimizer34 optimizes the transition graph generated bytransition graph generator32 in order to determine an optimal path of the transition graph. The optimal path represents an optimal price schedule.Forecast correction module35 determines a projected number of unrealized sales, and converts a demand forecast into a sales forecast.Elasticity module36 may be used to determine a projected price elasticity curve that transitiongraph generator32 may use to generate a transition graph.
• Adatabase40 stores data that may be used byserver30.History42 may include data that may be used to generate a price schedule for a product, such as inventory history or sales history of one or more products.History42 may include, for example, a number of units of a product sold, a price per unit, and a time at which the units were sold.Elasticity curve44 describes an estimated change in demand for a product in response to a change of a price of the product.Demand forecast43 describes a forecasted demand over time.Transition graph generator32 useselasticity curve44 to determine an estimated change in the demand of a product in response to a change in the price of the product.
• Locations24 and/ordatabase40 may be coupled toserver30 using one or more local area networks (LANs), metropolitan area networks (MANs), wide area networks (WANs), a global computer network such as the Internet, or any other appropriate wire line, wireless, or other links.
• FIG. 2 illustrates anexample transition graph50 that represents possible price schedules.Transition graph50 includesstates52 atdifferent stages60.Stages60 represent time intervals during which a product has a particular price.Stages60 may represent any suitable time interval, for example, a week. Additionally, the time intervals represented bystages60 may be equal to each other in length or may differ from each other, for example, using telescoping buffers. In the illustrated example, n stages60 are illustrated. A predecessor state may be coupled to a successor state by atransition53. A sequence ofstates52 from stage160ato stagen60ncoupled bytransitions53 form a path oftransition graph50.
• Astate52 may include any combination of values, for example, astage value51, aprice value54, aninventory value56, astate value58, and astate history59.Stage value51 represents the number of thestage60 at whichstate52 occurs.Price value54 represents the price per unit of a product.Price value54 may alternatively be associated with atransition53 from a predecessor state to a successor state.Inventory value56 represents the number of units of the product in the inventory ofcompany20, typically at the beginning of the state following transactions at previous states.
• State value58 represents the value ofstate52, and is used to calculate the value of the paths oftransition graph50.State value58 may be computed from, for example, the sales of a product, the expected profit of a product, the gross margin of a product, the inventory carrying cost of an optimal path leading to the state, the number of price changes of an optimal path leading to the state, or the average selling price of a product.State history59 ofstate52 includes information associated with a path from the initial state tostate52. For example,state history59 may include a number of price changes sincestage60aor a number of stages since a last price change.State history59 may include a certainty value that measures the certainty ofstate value58. The certainty value may be determined from an error margin associated with the calculation ofstate value58.
• In the illustrated example,transition graph50 includesstate52aatstage60a.State52ahas aprice value54 of “$100.00,” aninventory value56 of “500 units”, astate value58 of “0” (sincestate52ais an initial state), and a state history59h.State52ais a predecessor state to successor states52b-e. Successor states52b-erepresent possible states atstage60bgivenstate52aatstage60a.States52b-einclude price values “$100.00,” “$95.00,” “$90.00,” and “$85.00,” respectively. These price values54 represent possible prices of the product atstage60bgiven that the product has a price value of $100.00 atstage60a. Inventory values56 of “400 units” forstates52b-erepresent the number of units atstage60bgiven that there were five hundred units with a price value of $100.00 per unit atstage60a.Inventory value56 may be computed using a demand forecast, andelasticity curve44 provided byelasticity module36. State values58 ofstates52b-emeasure the value ofstates52b-egiven that one hundred units were sold at a price value of $100.00 per unit.
• Thevalues54,56,58, and59 may be quantized to reduce the number of successor states52, which may reduce the complexity oftransition graph50. Reducing the complexity oftransition graph50 may in turn reduce the processing power and time needed to generatetransition graph50, which may be important iftransition graph50 is large. In one embodiment,price value54 may be quantized. For example,specific price values54 may be predetermined for each stage. “Each” as used in this document means each member of a set or each member of a subset of a set. In the illustrated example, the price values “$100.00”, “$95.00,” “$90.00,” and “$80.00” may be predetermined forstates52b-e. Alternatively, changes in the price values54 from onestage60 to anotherstage60 may be predetermined. For example, price changes of $0.00, −$5.00, −$10.00, and −$15.00 may be predetermined forstates52b-e. Alternatively, price increments may be predetermined. For example, price values at $5.00 increments may be predetermined forstates52b-e.
• Similarly, inventory values56 may be quantized. Inventory increments may be predetermined, andinventory values56 at asuccessor stage60 computed fromprice value54 and aninventory value56 at apredecessor stage60 may be rounded up or down to the appropriatequantized inventory value56. For example, an inventory increment may be predetermined to be ten units, andinventory values56 ofstates52 may be rounded up or down to inventory values at increments of ten units. Rounding of inventory values, however, is not required to store the states, but may be used to compare states.
• Additionally, constraints may be used to reduce the number of successor states52 generated from apredecessor state52. Constraints may be placed onprice value54. For example, a maximum and a minimum price change may be used to limit the difference betweenprice values54 of asuccessor state52 and the price values54 of apredecessor state52. In the illustrated example, a maximum price change is −$15.00, and no minimum price change is given. An absoluteminimum price value54 and an absolutemaximum price value54 may be defined to limit theprice value54 of anystate52. In the illustrated example, a maximum price value is the starting price of “$100.00,” and a minimum price value is “$50.00.”
• A minimum and a maximum number of price changes may be used to control the number oftimes price value54 may change fromstage60ato stage60n. Alternatively, initial maximum and minimum price changes may be used to limit the price change fromstage60ato stage60b, and subsequent maximum and minimum price changes may be used to limit price changes fromstage60bto stage60n. A minimum number ofstages60 between price changes may be used to avoid frequent price changes. A maximum number of price changes may be defined to limit the number of price changes fromstage60ato stage60n.
• Constraints may be placed on inventory values56. For example, a maximum ending inventory may be used to eliminatestates52 atstage60nthat includeinventory values56 that are too high. Additionally, constraints may define a penalty or cost for inventory left at thelast stage60. Maximum and minimum inventory values56 may be used to control the unit's inventory at eachstage52.
• FIG. 3 illustrates an example method for generating an optimized price schedule for a product. Atstep78,elasticity module36 accesses demand forecast43 fromdatabase40. Atstep80,elasticity module36 computes anoptimal elasticity curve44.Elasticity curve44 describes the change in the demand for a product in response to a change in the price of the product. A method for computingelasticity curve44 is described in more detail with reference toFIG. 5.Demand forecast43 is adjusted atstep82.Demand forecast43 may be adjusted in order to take into account unrealized sales that are estimated at one or more sales locations24. A method for adjustingdemand forecast43 is described in more detail with reference toFIG. 7.Demand forecast43 is converted to a sales forecast atstep84. Generating an optimized price schedule for locations24 in the aggregate uses a sales forecast. A method for convertingdemand forecast43 is described in more detail with reference toFIG. 7.
• The values describing the product are quantized atstep86 to reduce the complexity of the computation. Atstep88,transition graph generator32 generates atransition graph50 that represents possible price schedules for a product. A method for generatingtransition graph50 described with reference toFIG. 4. Atstep90,optimizer34 determines an optimal path oftransition graph50. The optimal path represents an optimized price schedule forcompany20. Determining an optimal path is described in more detail with reference toFIG. 4. A price schedule is determined from the optimal path atstep92. The price schedule is output atstep94. Acomputer26 may output the price schedule.
• FIG. 4 illustrates an example method for generating an optimized price schedule using dynamic programming. Using dynamic programming techniques, atransition graph50 is generated. An optimal path oftransition graph50 is selected to determine an optimized price schedule. In one embodiment, a user at location24 may usecomputer26 to initiatetransition graph generator32, which may in turnuse elasticity module36 andoptimizer34 to generate an optimized price schedule.
• Generating a price schedule includes generatingtransition graph50 that comprises paths. Each path includesstates52, and eachstate52 may include, but is not limited to, the following: astage60 relating to the state's position in the time horizon, an assigned price, a projected inventory level, a state value, and a record of historic information, such as number of previous price changes. The inventory level, value, and history ofstate52 reflect the best path from the origin oftransition graph50 that leads tostate52.
• Transition graph50 may be generated by repeating the following for a number ofstages60 until a final stage is reached. The price value of a successor of acurrent state52 is determined using a forecast of product demand at thecurrent stage60 and a mathematical model of the current stage's price-demand response, also referred to as “price-demand elasticity.” The process of generating successor states may be called “state expansion”. Thestage60, inventory level, value, and history of thesuccessor state52 are calculated using the current state information in addition to the forecasted demand and price-demand elasticity for the current state. The process of state expansion may be carried out forstates52 of a given stage, advancingstage60 bystage60 through the time horizon in order to constructtransition graph50. After thefinal stage60 has been reached and states52 have been expanded, an optimal path is selected according to the state values and other information ofstates52. A price schedule is determined from the optimal path.
• Atstep150,transition graph generator32 definesstates52 and transitions53 oftransition graph50. States q_imay be defined by Equation (6):
  q_i:=(s_i,p_i,c_i,v_i,h_i)  (6)
• where:
• s_irepresents the stage,
• p_irepresents the price at stage s_i,
• c_irepresents the inventory of state q_i
• v_irepresents the value of state q_i,
• h_irepresents accumulated state history of state q_i.
• Set Q is defined as the set of all states q_i. While the state history may include various aspects relating to an optimal path to the state such as a number of price changes, the state representation as expressed by Equation (6) abstracts away the details of the state history. Different representations of a state may be used. Unless otherwise noted, the components of a state are labeled using the state's annotation. For example, the state q′ has components labeled (s′, p′, c′, v′, h′).
• State value v_iis used to evaluate states q_iduring state expansion. State value v_imay represent, for example, revenues accumulated through state q_i. Additional or alternative criteria may be used to select an optimal final state fromtransition graph50. For example, an evaluation function finalval: Q→R that maps states to real numbers may be used to evaluate inventory values and other values of a final state. A feasibility function finalfeasible: Q→{0,I} may be used to determine whether a final state is feasible according to the constraints. For example, a feasibility function may be used to determine whether a final inventory value is acceptable.
• Initial conditions and data are received atstep152. Initial conditions may include a maximum number of stages n, an initial state q₀, and a maximum number M of inventory units allowed at the last stage n. Other constraints may also be included in the initial conditions. Initial conditions may also include quantization criteria such as quantized inventory levels C and quantized price levels P. Quantization of inventory and price values may be defined by Equations (7):
  γ_c:R→C quantization of inventory  (7)
  γ_c:R→P quantization of price
• where R is the set of real numbers. In general, the quantization may be used to limit the number of states that may be generated at each stage while constructing the transition graph.
• Initial data may include a demand forecast F_sat a stage s, and an elasticity adjustment E(p,s) for price p at stage s that may be used to adjust demand forecast F_s. Elasticity adjustment E(p,s) may be determined from the elasticity curve calculated byelasticity module36 according to the method as described in connection withFIG. 5. For example, an elasticity constant c_Emay be extracted from the elasticity curve, as expressed by Equation (1), and may be used to determine E(p,s). If only the effect of price on demand is considered, and if elasticity is assumed to be constant over time and given according to the model in Equation (1), E(p,s) may be defined using Equation (8):$\begin{array}{cc}E\left(p,s\right)={\left(\frac{p}{p_0}\right)}^{c_E}& \left(8\right)\end{array}$
• Where
• c_E=elasticity constant
• p₀=base price on which forecast F_sis based
• Transition53 from apredecessor state52 to asuccessor state52 may be defined by a state-transition function expressed by Definition (9) that has a state-transition operator

  

  :
  (s,p,c,v,h)

  

  (s′,p′,c′,v′,h′), if and only if  (9)
• • s′=s+1,
  • p′εP,
  • c′=γ_c(max(0,c−F_s·E(p,s))),
  • v′=v+p·(c−c′),
  • h′captures the new state history and modifies h according to the transition, and
  • the transition to the new state is consistent with any additional constraints that may be imposed.
• Definition (9) provides an example of a state-transition function.Transition53, however, may be defined in any suitable manner. In addition, constraints may be placed ontransition53 in order to limit the successor states52 of astate52. Any of the components ofstate52 may be constrained. A constraint may, for example, prohibit drastic or frequent changes in the price values. A price-adjusted demand forecast F_s. E(p,s) may be adjusted to take into account unrealized sales that occur at one or more sales locations24. A method for adjusting the price-adjusted demand forecast is described in more detail in connection withFIG. 7.
• The following summary outlines a method for determining an optimal final state. The method expandstransition graph50 and determines an optimal feasible final state. An optimized price schedule may be determined from the optimal feasible final state andtransition graph50. The subsequent description provides a more detailed description of the method.

  Optimal Price Scheduling Algorithm (Summary)

  Input: Stages n, Forecast Fs,Elasticity E(p,s), Quantizations

  Initial staTe q0

  1	$K:=\left\{q_0\right\},U:=Q\text{❘}K,F:=\left\{q\in U:{\mathrm{<figure-callout\; label="q"\; filenames="US20080040202A1-20080214-D00002.png,US20080040202A1-20080214-D00005.png"\; state="\{\{state\}\}">q</figure-callout>}}_0\mapsto q\right\}$
  2	for k = 1 to n

  3	Sk:= {}
  4	$\mathrm{for}\mathrm{each}\stackrel{\_}{p}\in P$

  $S^k:=S^k\bigcup \{\left(q^{*},q^{\prime }\right)\in K⨯F:\stackrel{\_}{q}\in F\&$

  5

  $\left(q^{*},q^{\prime }\right)=\underset{\left({\mathrm{<figure-callout\; label="q"\; filenames="US20080040202A1-20080214-D00001.png,US20080040202A1-20080214-D00002.png"\; state="\{\{state\}\}">q</figure-callout>}}_1{q}_2\right)\in K⨯F}{\arg \max }\{v_2:{\mathrm{<figure-callout\; label="q"\; filenames="US20080040202A1-20080214-D00001.png,US20080040202A1-20080214-D00002.png"\; state="\{\{state\}\}">q</figure-callout>}}_1\mapsto {\mathrm{<figure-callout\; label="q"\; filenames="US20080040202A1-20080214-D00002.png"\; state="\{\{state\}\}">q</figure-callout>}}_2\&$

  ${\mathrm{<figure-callout\; label="q"\; filenames="US20080040202A1-20080214-D00002.png"\; state="\{\{state\}\}">q</figure-callout>}}_2=\left(k+1,\stackrel{\_}{p},{\mathrm{<figure-callout\; label="c"\; filenames="US20080040202A1-20080214-D00002.png"\; state="\{\{state\}\}">c</figure-callout>}}_2,{\mathrm{<figure-callout\; label="v"\; filenames="US20080040202A1-20080214-D00002.png"\; state="\{\{state\}\}">v</figure-callout>}}_2,\stackrel{\_}{h}\right)\&{\gamma }_c\left(c_2\right)={\gamma }_c\left(\stackrel{\_}{c}\right)\}$

  6	end
  7	K := K ∪ F
  8	U := U/F
  9	$F:=\left\{q^{\prime }\in U:\exists q\in K:q\mapsto q^{\prime }\right\}$
  10	end

  Output: argmaxq∈K:finalfensible(q){finalval(q)},

  	$\mathrm{Graph}\left(K,\bigcup _{k=1,\dots ,n}{S}^k\right)$

• Atstep154, data structures that describetransition graph50 are defined and initialized. The data structures may include set K at stage i, which is a set of maximal states q that may be coupled bytransitions53 from initial state q₀to a state at stage i. Amaximal state52 may be calculated according to state value v. Equation (10) describes a valid invariant for the states in set K:$\begin{array}{cc}\begin{array}{c}\forall q=\left(s,p,c,v,h\right)\in K\text{:}q\\ =\underset{q^{\prime }\in Q}{\arg \max }\left\{v^{\prime }\text{:}{\mathrm{<figure-callout\; label="q"\; filenames="US20080040202A1-20080214-D00002.png,US20080040202A1-20080214-D00005.png"\; state="\{\{state\}\}">q</figure-callout>}}_0\mapsto *q^{\prime }\&q^{\prime }=\left(s,p,c^{\prime },v^{\prime },h\right)\&{\gamma }_c\left(c^{\prime }\right)={\gamma }_c\left(c\right)\right\}\end{array}& \left(10\right)\end{array}$
• where

  

  denotes a possibly repeated application of the state-transition operator

  

  , as defined by Definition (9). Equation (10) expresses that at any time, set K contains only maximal states, that is, states (s, p, c, v, h) for which there is no better state, having the same values of s, p, h, and the same quantized value of c, that is reachable from q₀.
• Initially, set K is defined as {q₀}. Set F is a set of successor states52 that may be coupled to amaximal predecessor state52 of set K. Set U is a set ofunexplored states52 at astage60, and may be defined by U=(Q\K)\F. Set U may be initialized as U:=Q\K. Set S is a set oftransitions53 that couple a maximal predecessor state of set K to anunexplored successor state52 of set U. Set S maybe initialized as S:={ }.
• Successor states52 are generated atstep158. Successor states52 may be generated by adding state transitions to the set S^k, according to Equation (11):$\begin{array}{cc}{S}^k:=S^k\bigcup \{\left(q^{*},q^{\prime }\right)\in K\times F|\stackrel{\_}{q}\in F\&\left(q^{*},q^{\prime }\right)=\underset{\left({\mathrm{<figure-callout\; label="q"\; filenames="US20080040202A1-20080214-D00001.png,US20080040202A1-20080214-D00002.png"\; state="\{\{state\}\}">q</figure-callout>}}_1,q_2\right)\in K\times F}{\arg \max }\left\{v_z|{\mathrm{<figure-callout\; label="q"\; filenames="US20080040202A1-20080214-D00001.png,US20080040202A1-20080214-D00002.png"\; state="\{\{state\}\}">q</figure-callout>}}_1\mapsto {\mathrm{<figure-callout\; label="q"\; filenames="US20080040202A1-20080214-D00002.png"\; state="\{\{state\}\}">q</figure-callout>}}_2\&{\mathrm{<figure-callout\; label="q"\; filenames="US20080040202A1-20080214-D00002.png"\; state="\{\{state\}\}">q</figure-callout>}}_2=\left(k+1,\stackrel{\_}{p},{\mathrm{<figure-callout\; label="c"\; filenames="US20080040202A1-20080214-D00002.png"\; state="\{\{state\}\}">c</figure-callout>}}_2,{\mathrm{<figure-callout\; label="v"\; filenames="US20080040202A1-20080214-D00002.png"\; state="\{\{state\}\}">v</figure-callout>}}_2,\stackrel{\_}{h}\right)\&{\gamma }_c\left(c_2\right)={\gamma }_c\left(\stackrel{\_}{c}\right)\right\}& \left(11\right)\end{array}$
• Ordered pair (q*,q′) is a state-transition to be added, andq is a reachable state of set F that conforms to the specific price valuep. The argmax expression yields an ordered pair (q*,q′). State q′ of the ordered pair represents a value-maximal successor state q′, given fixed price valuep, state history valueh, and fixed inventory valuec, modulo the given quantization γ_c, and state q* represents the origin of the transition. Stateq binds inventory valuec and state history valueh, and enforces that only states which equivalent state history values are compared in the argmax expression. Set F, set K, and set U, are updated according to Equations (12):
  F:={q′εU:∃qεK:q

  

  q′}
  K:=K∪F
  U:=U\F  (12)
• Asuccessor state52 may be indexed by itsmaximal predecessor state52 in order to allow for efficient retrieval ofmaximal states52 fromprevious stages60. The index may be implemented in any suitable manner, for example, using a hash function or static array indexing. Invalid successor states52 may be eliminated according to the constraints specified in the initial conditions.
• Values for thesuccessor state52 are quantized and adjusted atstep160. Values for successor states52 may be quantized and adjusted using Equations (7). Price value p′ is quantized according to the quantization criteria specified in the initial conditions. Quantizing values reduces the complexity oftransition graph50, which may improve the processing time required to determine an optimal path oftransition graph50. Inventory c′ is also adjusted using the elasticity adjustment E(p,s). Inventory c′ may also be adjusted in order to take into account unrealized sales that occur at one or more sales locations24. A method for adjustinginventory values56 is described in more detail in connection withFIG. 7.
• Quantization may be applied in any suitable manner. For example, price values may be stored as quantized values, and inventory values may be stored as exact precise values. When different states are compared using Equation (11), the inventory values may be compared after quantization. If the quantized values are the same, the two successors can be regarded identical according to Equation (11).
• Atstep164,transition graph generator32 determines whether there is anext stage60. If there is anext stage60,transition graph generator32 returns to step156 to define successor states52 at thenext stage60. If there is nonext stage60 atstep164,transition graph generator32 proceeds to step166, whereoptimizer34 determines an optimal path oftransition graph50. The optimal path may be determined the at the final stage, which may be described by Equation (14), in terms of an optimal feasible solution and a transition graph:$\begin{array}{cc}\arg {\max }_{q\in K\text{:}\mathrm{finalfeasible}\left(q\right)}\left\{\mathrm{finalval}\left(q\right)\right\},\mathrm{Graph}\left(K,\bigcup _{k=1,\dots ,q}{S}^k\right)& \left(12\right)\end{array}$
• The selection of the optimal path may be limited by a constraint specified in the initial conditions, which may be labeled as finalfeasible. Additionally or alternatively, a predetermined certainty value may limit the optimal paths to paths that satisfy the predetermined certainty value.
• A price schedule is determined from the optimal path atstep168. The path comprises a sequence ofstates52 that describe a price value at eachstage60. The price values at eachstage60 are converted into a price schedule for a product. After the price schedule is determined, the method is terminated. A user may programtransition graph generator32 to periodically perform the method, for example, daily or weekly, in order to provide an updated price schedule generated using recent data.
• In one embodiment, the present invention may be used to compute an optimal sequence of product markdowns for liquidating merchandise, which may enable sellers to better plan and more profitably manage inventory, for example, at the end of a product lifecycle. Given initial inventory and price values, a price schedule of product markdowns that maximizes business objectives while satisfying business constraints may be calculated.
• FIG. 5 illustrates an example method for computing elasticity. In one embodiment, a user at location24 may usecomputer26 to initiateelasticity module36 to compute anelasticity curve44. According to the method,elasticity module36 determines an elasticity curve from state history and may use different filter sets to determine the elasticity curve. In general, each filter set is used to filter product data. Regression analysis is performed on the filtered data to yield elasticity curves. The fit of the data and the elasticity curves is measured, and an elasticity curve is selected.
• Elasticity module36 begins atstep110, where a demand model having dependent and independent variables is determined. The demand model may be defined by, for example, Equation (1):
  D=KP^CE(1+a)^qt^βeⁿ  (1)
• where:
• D=demand
• P=price
• α=promotional variable
• t=time
• c_E=elasticity constant
• K, q, β, and γ are constants, K>O, and it is expected that c_E<0, β>0, and γ<0. Equation (1) may be rewritten as Equation (2):
  ln(D)=ln(K)+cln(P)+qln(1+α)+β ln(t)+γt  (2)
• Equation (2) may be used as a regression equation, where ln(D) is the dependent variable and ln(P), ln(1+α), ln(t), and t are the independent variables.
• Atstep112, values for the dependent and independent variables are received. The values may be derived from data describing one or more products. The products may include the product for which a price schedule is to be generated and/or products similar to the product for which a price schedule is to be generated. For example, a single product may not have sufficient sales history data to generate a meaningful price schedule, so data from similar products may be combined to generate a price schedule.
• Data for a product may be grouped in data rows, where each data row includes data for a specific stage in time. A data row may include, for example, the following values: Product Identifier, Stage Number, Sales, Price, Revenues, and Promo. Product Identifier identifies the product, and Stage Number identifies the stage. Sales describes the number of units of the products sold during the stage, Price defines the price per unit, and Revenues describe the revenues generated from the sale of the product. Promo describes the number of units sold during a promotion for the product, such as a temporary price reduction. Other or additional values may be used to calculate an elasticity curve. For example, a seasonal index variable that describes how a time of year, such as the Christmas season, affects the demand for a product maybe used.
• The values for Sales may be used as the values for demand D. Values for Price and Stage Number may be used as the values for price P and time t. Promotion variable α measures the significance of a promotion, and may be defined using Equation (3):$\begin{array}{cc}\alpha :=\{\begin{array}{cc}\frac{\mathrm{Promo}}{\mathrm{Sales}},& \mathrm{if}\mathrm{Promo}>\mathrm{Promo}\mathrm{<figure-callout\; label="Filter"\; filenames="US20080040202A1-20080214-D00002.png,US20080040202A1-20080214-D00005.png"\; state="\{\{state\}\}">Filter</figure-callout>}\\ 0& \mathrm{otherwise}\end{array}& \left(3\right)\end{array}$
• where Promo Filter is a constraint.
• Filter sets are defined atstep114. Each filter set comprises one or more constraints that restrict the values for the variables of the demand model. An example filter set is described by Table 1:

  	TABLE 1
  			Example
  	Filter Name	Description	Value
  	Min Sales	Minimum number ofsales	5 units
  	Min Revenues	Minimum revenues	$10
  	Min Rows	Minimum number of data rows	15 points
  	Promo Filter	Promotions filter	5%
  	Perc Filter	Data value filter	2.5%

• According to the filter set of Table 1, a data row is removed if Sales is less than Min Sales or Revenues is less than Min Revenues. The data rows for a product are removed if number of data rows for the product is less than Min Rows. Promo Filter is used to define promotion variable α, as shown in Equation (3). Data rows that include values that are within a first or last percentile given by Perc Filter are removed.
• Steps116 through144 are performed for each filter set in order to determine the filter set that yields an optimal elasticity curve. Atstep116, a filter set is selected.Steps118 through136 are performed on data for each product individually in order to eliminate unacceptable data. Atstep118, a product is selected. Atstep120, the values for the product are filtered using the filter set selected atstep116. The variables are normalized atstep122. Variables may be normalized according to Equations (4):
  Norm(D)=ln(D)−Mean(ln(D))
  Norm(P)=ln(P)−Mean(ln(P))
  Norm(t)=t−Mean(t)
  Norm(ln(t))=ln(t)−Mean(ln(t))
• where Mean (x) is the average of x.
• Atstep124, a regression analysis is performed using the demand model, as expressed by Equation (2), as the regression equation of the dependent variable over the independent variables, that is, of ln(D) over ln(P), ln(1+α), ln(t), and t. The regression analysis yields values for coefficients c_E, K, q, β, and γ, which are inserted into Equation (2) to define a regression or elasticity curve. The quality of the elasticity curve is measured atstep126. The quality may be measured by determining the fit of the elasticity curve to the values that were used to generate the elasticity curve, and may be measured using a Box-Wetz value B expressed by Equation (5):$\begin{array}{cc}B=\left({\max }_i{\hat{Y}}_i-{\min }_i{\hat{Y}}_i\right)/\sqrt{\frac{p{s}^2}{n}})& \left(5\right)\end{array}$
• where:$p=\mathrm{number}\mathrm{of}\mathrm{parameters}$$s^2\approx {\mathrm{<figure-callout\; label="\sigma "\; filenames="US20080040202A1-20080214-D00002.png"\; state="\{\{state\}\}">\sigma </figure-callout>}}^{2}V\left(Y_i\right)$$\stackrel{\_}{V}\left(\hat{Y}\right)=\sum _{i}V\left({\hat{Y}}_i\right)/n$
• The quality of the elasticity curve is determined to be acceptable or not atstep128. The quality may be acceptable if it satisfies a predetermined criteria. If the quality is not acceptable, the data for the product is eliminated atstep130, andelasticity module36 proceeds to step138. If the quality is acceptable,elasticity module36 proceeds to step132.
• Atstep132, an elasticity value is determined from the elasticity curve. The negative of the elasticity coefficient c_Eof the price variable P may be used as the elasticity value. In one embodiment, the elasticity value is required to be positive.
• Atstep138,elasticity module36 determines whether there is a next product. If there is a next product,elasticity module36 returns to step118, where the next product is selected. If there is no next product,elasticity module36 proceeds to step140.
• Steps140 through144 are performed on the aggregated data for the products in order to determine which filter set provides an optimized fit of the elasticity curve. Atstep140, the data for the remaining products are aggregated to yield aggregated data. Atstep142, regression analysis is performed on the aggregated data. The regression analysis may use the demand model as expressed by Equation (2) as the regression equation of the dependent variable over the independent variables. The quality of the elasticity curve is measured atstep144, and may be measured in a manner substantially similar to that described above in connection withstep126.
• Atstep138,elasticity module36 determines whether there is a next filter set. If there is a next filter set,elasticity module36 returns to step116, where the next filter set is selected. If there is no next filter set,elasticity module36 proceeds to step148, where an optimal elasticity curve is selected. The optimal elasticity curve may be the highest quality elasticity curve according to measurements performed atstep144. Additionally, the elasticity curve may be rewritten in the format of Equation (1) and sent to transitiongraph generator32. After selecting the optimal elasticity curve,elasticity module36 terminates the method. A user may programelasticity module36 to perform the method periodically, for example, daily or weekly, in order to provide an updated elasticity curve to transitiongraph generator32.
• FIGS. 6A through 6C illustrate anexample elasticity curve40.FIG. 6A illustrateselasticity curve40 with respect to time.Elasticity curve40 is computed from price values and demand values.FIG. 6B illustrateselasticity curve40 and the demand values with respect to time.FIG. 6C illustrateselasticity curve40 and the demand values with respect to the price values.
• FIG. 7 illustrates an example method for estimating a sales forecast from a demand forecast and an aggregate inventory value that describes the total inventory spread across a number of locations24 ofcompany20. A demand forecast typically predicts customer demand, but not sales. Computing an optimized price schedule for a number of locations24 in the aggregate, however, requires the prediction of sales instead of demand. An aggregate sales forecast may be calculated from the aggregate inventory value and demand values from predecessor states52.
• Unrealized sales, or stockouts, may be determined in order to predict sales. An unrealized sale is predicted where there is a demand at a location, but no inventory. An unrealized sale may be expressed as a difference between a demand value and an inventory value at any given location24. According to the method, a demand forecast is adjusted to take into account unrealized sales that may be predicted at one or more locations24 ofcompany20, which may improve the accuracy of the predicted inventory values of an optimized price schedule.
• The method performs a significant adjustment to the demand forecast when inventory levels and forecasted demands are aggregated across a number of locations24, or a store-group. The method may be used with the method for generating an optimized price schedule as described with reference toFIG. 4, in order to improve the accuracy of the predicted number of sales when operating on a store-group level. The method may be used at each state-expansion in order to estimate the unit sales from the demand forecast, thereby significantly improving the accuracy of the inventory level of a successor state, in particular at low inventory levels. The method may be calibrated to a particular location24 and replenishment situation based on historic sales and inventory figures.
• This method may be applied as part of the computation of the optimal price schedule, particularly if the estimate is not accurately performed as a preprocessing step. This may be the case if the demand value at each stage of the price scheduling horizon is dependent on a price value, and the future inventory values are dependent on the demand value. As has been outlined above, the prediction of sales in turn depends on the inventory values, leading to a circular dependency. The circular dependency may be resolved by taking the effect of distributed inventory on projected sales into account at each step of the computation of a price schedule.FIG. 7 describes how a projected number of sales can be computed, given a demand forecast and an aggregate inventory at locations24.
• Atstep180,transition graph generator32 defines a number n of locations24. An inventory s at each location24 is estimated at step182. Inventory s may be estimated by, for example, randomly populating each location24 with inventory units. A demand at d at each location24 is estimated atstep184. Demand d at each location24 may be estimated by randomly populating each location24 with demand units. This random sampling may be performed a number of times. The expected number E(M) of unrealized sales is determined atstep186.
• Alternatively, steps182 through186 may be performed using a probabilistic estimation. Given S units that are assumed to be evenly distributed across a number of n locations, the probability of a location receiving s units may be expressed using a binomial distribution. The binomial distribution may be used to determine the probability of s successful experiments, where the probability of success is 1/n. The random variable that represents the number of units of inventory at any particular store may be denoted by X_s. At step182, inventory s at each location24 may be estimated by calculating the probability of a location24 having inventory s using Equation (15):$\begin{array}{cc}p\left(X_S=s\right)=\mathrm{Bnp}\left(S,s,\frac{1}{n}\right)=\left(\begin{array}{c}S\\ s\end{array}\right)·{\left(\frac{1}{n}\right)}^s·{\left(1-\frac{1}{n}\right)}^{S-s}& \left(15\right)\end{array}$
• where S represents the total amount of inventory at all locations24. Similarly, atstep184, the probability of demand d at each location24 may be calculated using Equation (16):$\begin{array}{cc}p\left(X_D=d\right)=\mathrm{Bnp}\left(D,d,\frac{1}{n}\right)=\left(\begin{array}{c}D\\ d\end{array}\right)·{\left(\frac{1}{n}\right)}^d·{\left(1-\frac{1}{n}\right)}^{D-d}& \left(16\right)\end{array}$
• where D represents the aggregate demand summed over locations24, and X_Ddenotes the random variable that represents the unit demand at a particular location24.
• The expected number of unrealized sales E(M) may be calculated atstep186 using Equation (17):$\begin{array}{cc}\begin{array}{c}E\left(M\right)=\sum _{s=1}^S\sum _{d=s+1}^{D}p\left(S=s,D=d\right)·\left(d-s\right)\\ =\sum _{s=1}^{S}p\left(S=s\right)\sum _{d=s+1}^{D}p\left(D=d\right)·\left(d-s\right)\\ \sum _{s=1}^S\mathrm{Bnp}\left(S,s,\frac{1}{n}\right)·\sum _{d=s+1}^D\mathrm{Bnp}\left(D,d,\frac{1}{n}\right)\left(d-s\right)\\ =\sum _{s=1}^S[\left(\begin{array}{c}S\\ s\end{array}\right)·{\left(\frac{1}{n}\right)}^s·{\left(1-\frac{1}{n}\right)}^{S-s}·\\ \sum _{d=s+1}^D\left(\begin{array}{c}D\\ d\end{array}\right)·{\left(\frac{1}{n}\right)}^d·{\left(1-\frac{1}{n}\right)}^{D-d}\left(d-s\right)]\end{array}& \left(17\right)\end{array}$
• To efficiently implement computation of unrealized sales E(M) onsystem10, different numerical approximations schemes may be used, depending on the value ranges of the input parameters. In the following, several approximation schemes are described that can be applied according to the value ranges of D, S, and n.
• The binomial distribution as expressed by Equation (18):$\begin{array}{cc}\mathrm{Bnp}\left(k,n,p\right)=\left(\begin{array}{c}n\\ k\end{array}\right){p^k\left(1-p\right)}^{n-k}& \left(18\right)\end{array}$
• may be approximated by the Poisson distribution for a sufficiently large n, a sufficiently small p, and moderately sized np, as expressed by Equation (19):$\begin{array}{cc}\mathrm{Bnp}\left(k,n,p\right)\approx \frac{{\lambda }^k{e}^{-\lambda }}{k!},& \left(19\right)\end{array}$
• where λ=np. This can be applied, for example, to the computation of E(M) at step182, if np≧10. Thus, steps182 through186 may be performed using a Poisson approximation to a binomial distribution by computing the probability of a location24 having an inventory s or demand d using Equation (19).
• Atstep186, expected number E(M) may be determined using Equation (20):$\begin{array}{cc}\begin{array}{c}E\left(M\right)=\sum _{s=1}^S\sum _{d=s+1}^D\mathrm{Bnp}\left(S,s,\frac{1}{n}\right)·\mathrm{Bnp}\left(D,d,\frac{1}{n}\right)\left(d-s\right)\\ =\sum _{s=1}^S\left[\frac{{\left({\lambda }_S\right)}^s{e}^{-{\lambda }_s}}{s!}·\sum _{d=s+1}^D\frac{{\left({\lambda }_D\right)}^d{e}^{-{\lambda }_D}}{d!}\left(d-s\right)\right]\end{array}& \left(20\right)\end{array}$
• Alternatively, the computation in Equation (17) may be performed by using an incomplete beta-function. The inner sum of Equation (17) may be denoted by J, and Equations (21) through (28) may be used to describe the steps to transform J into a form that may be efficiently computed using a standard procedure for the incomplete beta-function.$\begin{array}{cc}\begin{array}{c}J=\sum _{d=s+1}^D\mathrm{Bnp}\left(D,d,\frac{1}{n}\right)\left(d-s\right)\\ =\sum _{d=s+1}^D\left(\begin{array}{c}D\\ d\end{array}\right){\left(\frac{1}{n}\right)}^d{\left(1-\frac{1}{n}\right)}^{D-d}\left(d-s\right)\end{array}& \left(21\right)\end{array}$
• Probabilities p and q are defined using Equations (22):$\begin{array}{cc}p=\frac{1}{n}q=1-p& \left(22\right)\end{array}$
• J may then be expressed by J1 and J2, as shown by Equation (23):$\begin{array}{cc}\begin{array}{c}J=\sum _{d=s+1}^D\left(\begin{array}{c}D\\ d\end{array}\right)p^d{q}^{D-d}d-\sum _{d=s+1}^D\left(\begin{array}{c}D\\ d\end{array}\right)p^d{q}^{D-d}s\\ =J1-J2\end{array}\mathrm{where}J1=\sum _{d=s+1}^D\left(\begin{array}{c}D\\ d\end{array}\right)p^d{q}^{D-d}d,\mathrm{and}J2=s\sum _{d=s+1}^D\left(\begin{array}{c}D\\ d\end{array}\right)p^d{q}^{D-d}& \left(23\right)\end{array}$
• Expression$\left(\begin{array}{c}D\\ d\end{array}\right)d$
  may be substituted by Equation (24):$\begin{array}{cc}\left(\begin{array}{c}D\\ d\end{array}\right)d=\frac{D!d}{\left(D-d\right)!d!}=D\frac{\left(D-1\right)!}{\left(D-d\right)!\left(d-1\right)!}=D\left(\begin{array}{c}D-1\\ d-1\end{array}\right)& \left(24\right)\end{array}$
• Furthermore, J1 may be described by Equation (25):$\begin{array}{cc}\begin{array}{c}J1=\mathrm{Dp}\sum _{d=s+1}^D\left(\begin{array}{c}D-1\\ d-1\end{array}\right)p^{d-1}{q}^{D-d}\\ =\mathrm{Dp}\sum _{d^{\prime }=s}^{D^{\prime }}\left(\begin{array}{c}{D}^{\prime }\\ d^{\prime }\end{array}\right)p^{d^{\prime }}{q}^{D^{\prime }-d^{\prime }}\end{array}\mathrm{where}\text{:}{D}^{\prime }=D-1.& \left(25\right)\end{array}$
• To compute J, the components J1 and J2 and Equation (26) may be used:$\begin{array}{cc}\sum _{l=k_1}^Q\left(\begin{array}{c}Q\\ I\end{array}\right)p^l{q}^{Q-1}=\mathrm{Ip}\left({\mathrm{<figure-callout\; label="k"\; filenames="US20080040202A1-20080214-D00001.png,US20080040202A1-20080214-D00002.png"\; state="\{\{state\}\}">k</figure-callout>}}_1,Q-{\mathrm{<figure-callout\; label="k"\; filenames="US20080040202A1-20080214-D00001.png,US20080040202A1-20080214-D00002.png"\; state="\{\{state\}\}">k</figure-callout>}}_1+1\right)& \left(26\right)\end{array}$
• where I_x(a,b) is an incomplete beta function given by Equation (27),$\begin{array}{cc}{I}_x\left(a,b\right)=\frac{1}{B\left(a,b\right)}{\int }_0^x{t^{a-1}\left(1-t\right)}^{b-2}dt\mathrm{where}& \left(27\right)\\ B\left(a,b\right)={\int }_0^{\infty }{t^{a-1}\left(1-t\right)}^{b-1}dt& \left(28\right)\end{array}$
• Atstep188, the aggregate sales forecast is computed according to the expected number of unrealized sales. An aggregate sales forecast at asuccessor state52 is calculated by subtracting the expected sales from the aggregate inventory at apredecessor state52. The demand does not take into account unrealized sales. Thus, the expected number of unrealized sales may be subtracted from the forecasted demand to account for unrealized sales, which may yield a more accurate price schedule. After the forecast is adjusted, the method is terminated.
• The present invention may efficiently generate an optimized price schedule by using dynamic programming. In dynamic programming,transition graph50 with paths representing possible price schedules is generated, and an optimal path of thetransition graph50 is selected in order to determine an optimized price schedule. The present invention may also improve the efficiency of the optimization by using quantized dynamic programming, which involves quantifying values associated withtransition graph50. Quantizing the values reduces the complexity oftransition graph50, which may improve the processing time and power required to determine an optimal path oftransition graph50. In addition, the present invention may use an improved calculation of elasticity in order to increase the accuracy of the optimization. Moreover, the present invention may take into account an unrealized sale that occurs at one of many locations, which may improve the accuracy of the optimal path.
• Although the present invention has been described with several embodiments, a myriad of changes, variations, alterations, transformations, and modifications may be suggested to one skilled in the art, and it is intended that the present invention encompass such changes, variations, alterations, transformations, and modifications as fall within the scope of the appended claims.

Claims (21)

1. A method for determining a sales forecast, comprising:

defining a plurality of locations;

estimating an inventory at each location;

estimating a demand at each location;

calculating an expected number of unrealized sales at each location using a difference between the demand at the location and the inventory at the location; and

determining a sales forecast in response to the expected number.

2. The method ofclaim 1, wherein:

estimating the inventory at each location comprises randomly populating the locations with a plurality of inventory units; and

estimating the demand at each location comprises randomly populating the locations with a plurality of demand units.

3. The method ofclaim 1, wherein:

estimating the inventory at each location comprises calculating a probability of each location receiving a number of inventory units according to a binomial distribution; and

estimating the demand at each location comprises calculating a probability of each location receiving a number of demand units according to the binomial distribution.

4. The method ofclaim 1, wherein:

estimating the inventory at each location comprises calculating a probability of each location receiving a number of inventory units according to an incomplete beta-function; and

estimating the demand at each location comprises calculating a probability of each location receiving a number of demand units according to the incomplete beta-function.

5. A system for determining a sales forecast, comprising:

a database operable to store a plurality of definitions defining a plurality of locations; and

a server coupled to the database and operable to:

estimate an inventory at each location;

estimate a demand at each location;

calculate an expected number of unrealized sales at each location using a difference between the demand at the location and the inventory at the location; and

determine a sales forecast in response to the expected number.

6. The system ofclaim 5, wherein the server is operable to:

estimate the inventory at each location by randomly populating the locations with a plurality of inventory units; and

estimate the demand at each location by randomly populating the locations with a plurality of demand units.

7. The system ofclaim 5, wherein the server is operable to:

estimate the inventory at each location by calculating a probability of each location receiving a number of inventory units according to a binomial distribution; and

estimate the demand at each location by calculating a probability of each location receiving a number of demand units according to the binomial distribution.

8. The system ofclaim 5, wherein the server is operable to:

estimate the inventory at each location by calculating a probability of each location receiving a number of inventory units according to an incomplete beta-function; and

estimate the demand at each location by calculating a probability of each location receiving a number of demand units according to the incomplete beta-function.

9. Logic for determining a sales forecast, the logic encoded in media and when executed operable to:

define a plurality of locations;

estimate an inventory at each location;

estimate a demand at each location;

calculate an expected number of unrealized sales at each location using a difference between the demand at the location and the inventory at the location; and

determine a sales forecast in response to the expected number.

10. The logic ofclaim 9, further operable to:

estimate the inventory at each location by randomly populating the locations with a plurality of inventory units; and

estimate the demand at each location by randomly populating the locations with a plurality of demand units.

11. The logic ofclaim 9, further operable to:

estimate the inventory at each location by calculating a probability of each location receiving a number of inventory units according to a binomial distribution; and

estimate the demand at each location by calculating a probability of each location receiving a number of demand units according to the binomial distribution.

12. The logic ofclaim 9, further operable to:

estimate the inventory at each location by calculating a probability of each location receiving a number of inventory units according to an incomplete beta-function; and

estimate the demand at each location by calculating a probability of each location receiving a number of demand units according to the incomplete beta-function.

13. A system for determining a sales forecast, comprising:

means for defining a plurality of locations;

means for estimating an inventory at each location; means for estimating a demand at each location;

means for calculating an expected number of unrealized sales at each location using a difference between the demand at the location and the inventory at the location; and

means for determining a sales forecast in response to the expected number.

14. A method for generating a price schedule, comprising:

generating a transition graph comprising a plurality of paths, each path comprising a plurality of states, each state having a plurality of values comprising a state value, the transition graph being generated by repeating the following for a plurality of stages until a final stage is reached;

calculating the values of a successor state using the values of a predecessor state; and

quantizing the values of each successor state;

selecting an optimal path according to the state values of the states; and

determining a price schedule from the optimal path.

15. The method ofclaim 14, wherein the values comprise a price value.

16. The method ofclaim 14, wherein the values comprise an inventory value.

17. The method ofclaim 14, wherein selecting the optimal path according to the state values comprises:

determining a state at the final stage having an optimal state value; and

determining a path comprising a state of an initial stage and the state having the optimal state value.

18. The method ofclaim 14, further comprising eliminating a successor state in response to a constraint.

19. The method ofclaim 14, further comprising:

computing an elasticity curve; and

computing an inventory value of each successor state using the elasticity curve.

20. The method ofclaim 14, wherein:

each state has a certainty value; and

selecting the optimal path comprises determining a state at the final stage having a certainty value of a predetermined value.

21. The method ofclaim 14, further comprising:

defining a plurality of locations;

calculating an expected number of unrealized sales at each location; and

adjusting a value of the successor state in response to the expected number.

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