Patent Application
20130166353
One embodiment is directed generally to a computer system, and in particular to a computer system that optimizes product pricing.
Product line pricing is an important business problem faced by retailers and other sellers of merchandise who employ dynamic pricing strategies to generate incremental revenue benefits throughout the year. Retailers, among others, have in increasing numbers begun to utilize decision support systems that leverage the large volume of detailed demand data to automate and optimize pricing recommendations. In particular, the statistical modeling of the price elasticity of items based on analyzing the effect of price changes of one product on its demand, or the demand for another product, can be used to optimize the pricing of products.
Known price optimizers, in general, are given a set of items in a product category together with their current prices and demand cross-elasticity. The optimizers then find a new set of prices that would satisfy all business rules and maximize the desired objective (e.g., profit margin, revenue, etc.) by taking into account some soft price constraints and allowed prices.
One embodiment is a price optimization system that determines the pricing of a plurality of items. The system receives an initial price vector for the items and assigns the initial price vector as a current price vector. The system further receives an objective function. The system determines a first new price vector by exploiting the current price vector, where the exploiting includes randomly choosing a first set of allowed prices for the items, and when the first set of allowed prices improves the objective function, assigning the first set of allowed prices as the current price vector. The system then determines a second new price vector by exploring the current price vector, where the exploring includes randomly choosing a second set of allowed prices for the items, and when the second set of allowed prices does not decrease the objective function by more than a predetermined value, assigning the second set of allowed prices as the current price vector. The system sequentially repeats the exploiting and exploring functionality until a terminating criteria is reached. When the terminating criteria is reached, the current price vector is the determined pricing of the plurality of items.
a and 3b is a flow diagram of the functionality of the product pricing module when determining a new price vector by executing the exploitation phase or the exploitation phase in accordance with one embodiment.
One embodiment is a product price optimizer that searches for optimal pricing using two sequential randomized search phases, an “exploration” phase and an “exploitation” phase. The phases alternate until the there is a convergence to an optimal solution or until a maximum run time is reached. Each phase consists of repetitive cycles where the items are considered in random order using uniform probability distribution. In the exploitation phase, new solutions are accepted if they improve an objective function such as maximal revenue or gross margin. In the exploration phase, new solutions are accepted if they do not decrease the previously found best objective value by more than a specified percentage.
One embodiment is directed to a price optimizer for setting the prices of items in each category carried by a large retail store so as to maximize an “objective function”, which in general depends on the sales volume of all items. The most common metric for the objective function is profit or revenue. When the prices of items are changed, the change in the sales volume of each item depends on the product of the percentage changes in the prices of all other items, with each percentage change usually raised to some power. Therefore, the objective function is a nonlinear function of the prices. Further, every feasible price vector (i.e., the determined prices of all items) must generally satisfy multiple constraints, some of which specify that the profit, revenue, and sales volume have to be greater than a certain specified fraction of the original profit, revenue and sales volume. Therefore, the constraints are also nonlinear.
In general, a Linear Programming (“LP”) methodology is the standard methodology for solving constrained optimization problems with linear constraints and a linear objective function. However, the nonlinearities present in many price optimization problems, as discussed above, require problem-specific approximations and linearizations to be made before the LP approach can be used, and a solution found using LP will necessarily be suboptimal. Further, the nonlinearity of constraints implies that the solution set is not convex, which means there may be multiple local maxima for the objective function, and hence nonlinear gradient search methods will in general not find the global maximum of the “price optimization problem”.
Many real-world optimization problems are formulated to minimize or maximize the nonlinear function of discrete variables subject to certain constraints involving that function or some linear functions of its arguments. An example of this problem type, the price optimization problem, is the maximization of the objective function of total revenue for several merchandise items in a category where demand for each item is expressed as a nonlinear function of all item prices, which can take values only from specific price ladders (i.e., a set of allowed prices). The price optimization problem can be stated as follows:
Find the best prices (p1, . . . , pn) for the n product items in a given merchandise category to maximize the total revenue subject to certain linear constraints when demand for the ith item is expressed as a given demand function di(p1, . . . , pn) (i.e., a demand model). The problem can also be expressed as follows:
where di(p1, . . . , pn) is a nonlinear expression for the ith item demand as a function of all prices in the category; Pi is the price ladder (discrete set) for the ith item; and Lp (Ld, Lr) are the feasible price (demand and revenue, respectively) regions defined by the linear inter-item constraints. Each item i has a cost associated with it.
In known product pricing optimizers that generate optimized prices, the optimization problem shown in equations (1-3) above is solved by a series of greedy searches, which sometimes delivers a solution far from optimal. Other known approaches are based on linear or quadratic approximations of the nonlinear demand function. However, these approaches tend to suffer from insufficient accuracy (linear approximation) or an overwhelming number of integer variables needed to formulate the problem.
In contrast with known LP approaches of optimizing prices, embodiments of the present invention use randomized search (“RS”) to arrive at an optimal or near optimal solution to the price optimization problem, despite its nonlinearities. In general, one embodiment is a price optimizer that calculates a set of recommended prices or validates a set of user-defined input prices, given one or more of the following inputs:
Computer readable media may be any available media that can be accessed by processor 22 and includes both volatile and nonvolatile media, removable and non-removable media, and communication media. Communication media may include computer readable instructions, data structures, program modules or other data in a modulated data signal such as a carrier wave or other transport mechanism and includes any information delivery media.
Processor 22 is further coupled via bus 12 to a display 24, such as a Liquid Crystal Display (“LCD”), for displaying information to a user. A keyboard 26 and a cursor control device 28, such as a computer mouse, is further coupled to bus 12 to enable a user to interface with system 10.
In one embodiment, memory 14 stores software modules that provide functionality when executed by processor 22. The modules include an operating system 15 that provides operating system functionality for system 10. The modules further include a product pricing module 16 that generates optimized product pricing using randomized search, as disclosed in more detail below. System 10 can be part of a larger system, such as an enterprise resource planning (“ERP”) system. Therefore, system 10 will typically include one or more additional functional modules 18 to include the additional functionality. A database 17 is coupled to bus 12 to provide centralized storage for modules 16 and 18 and store pricing information, inventory information, ERP data, etc.
In one embodiment, a revenue maximization problem is developed and solved for category pricing in the retail industry. However, in other embodiments this problem can be used for any type of product pricing. For the retail embodiment, consider a retailer who has to set the baseline (or regular) price levels for some or all active items in a given category for the next few months, as part of a merchandise planning process. The category manager has to make multiple, coordinated pricing decisions, proactively taking into account the impact of a price change on the sales of other items within the category, as well as any (extraneous) market response. Moreover, the recommended prices have to satisfy several category-level objectives such as profitability, sales, and revenue (e.g., to maximize gross margin while ensuring that the total sales and revenue are within 10% of the current value), and have to be selected from within a limited discrete price ladder (e.g., be within 20% of the current price and end with ‘9’ cents). In addition, items have to be priced relative to certain attributes such as brand type (e.g., a store brand tomato soup should be at least a dollar less than the price of the corresponding national brand), and quantity (e.g., a six-pack of diet-soda versus a two-liter bottle of diet-soda), among others.
Items can represent stock-keeping units (“SKU”s), product subclasses, or product classes within the category, depending on the level of aggregation in the merchandise hierarchy at which the analysis is performed by the category manager. For simplicity, it is assumed that prices of SKUs are optimized at the store-level of the location hierarchy. However, in other embodiments, the problem can be readily extended to manage higher levels of aggregation (e.g., at the zonal level). Further, the problem can address more general situations faced by category managers such as the need to jointly optimize multiple categories that are inter-linked by pricing constraints and/or objectives, or manage several distinct subsets of substitutable items within the same category.
As disclosed, one embodiment determines a set or vector of optimized prices that maximize an objective function and meets all constraints. The constraints can be expressed as follows:
In one embodiment, the objective function or value can be expressed as follows:
f
0
=W
v·Volume+Wr·Revenue+Wm·Margin
One embodiment searches for optimal pricing using the two sequential randomized search phases, the “exploration” phase and the “exploitation” phase. The phases alternate until the there is a convergence to an optimal solution or until a maximum run time is reached. Each phase consists of repetitive cycles where the items are considered in random order using uniform probability distribution. In the exploitation phase, new solutions are accepted if they improve an objective function such as maximal revenue or gross margin. In the exploration phase, new solutions are accepted if they do not decrease the previously found best objective function by more than a specified percentage.
At 202, the initial setting and constraints are received.
At 204, it is determined if an initial price vector is provided. If no, at 206 a feasible initial price/solution vector is found. A feasible initial price vector in one embodiment is an assigned price for all items of interest, where the price for each item is one of the discrete prices on a pricing ladder for each item. The initial price vector is the vector of currently used prices. If, for example, a user was attempting to find optimal prices for a brand new store with no current prices, an initial price vector can be chosen randomly and the functionality of module 16 will ultimately arrive at optimized pricing.
If yes at 204, at 210, a new price vector, “vnew” is found via the “exploitation” phase that exploits the solution space around the initial price vector. The exploitation phase portion of determining a new price vector, as disclosed in more detail in conjunction with
At 212, vcurrent=vnew.
At 214, new price vector, “vnew” is found via the “exploration” phase that explores the solution space around vcurrent. The exploration phase portion of determining a new price vector, as disclosed in more detail in conjunction with
At 216, vcurrent=vnew.
At 222, as in 210, a new price vector, “vnew” is found via the “exploitation” phase that exploits the solution space around vcurrent.
At 224, vcurrent=vnew.
At 218, it is determined if a terminating criteria is reached, for example a limit of how many cycles of steps 214, 216, 222 and 224 are executed without realizing improvements in the objective value. In one embodiment, if three cycles of 214, 216, 222 and 224 are executed without improvement, the functionality of
If the terminating criteria is reached at 218, the functionality of
a and 3b is a flow diagram of the functionality of product pricing module 16 when determining a new price vector by executing the exploitation phase or the exploitation phase in accordance with one embodiment.
At 301, the functionality of finding a new price vector is initiated.
At 302, the pool of available items, “So”, is constructed.
At 304, variables vo and vnew are set: vo=vcurrent and vnew=vcurrent.
At 306, the value of the objective function, “fcurrent”, is found at price vector vo. As disclosed above, in one embodiment the objective function is determined by f0=Wv·Volume+Wr·Revenue+Wm·Margin where the Volume is determined as follows:
The total revenue for each item “k” is the sum over all items k of the volumek*pricek, and the profit margin is the sum over all items k of volumek*(pricek−costk), where costk (i.e., the cost of each item) is known ahead of time.
At 308, it is determined if the functionality is the exploration phase (i.e., whether 301 of
At 330, an item “i” is chosen from So at random. [At 332, the set of allowed prices (i.e., prices from the price ladder, and taking into account inter-item constraints), Sa, for the item i is constructed. At 334, it is determined if Sa is empty. If there are no price levels in the price ladder for which the constraints are satisfied, then Sa is empty. The set Sa may be repeatedly revisited through the functionality of
If yes at 334, functionality continues at 366 where an item i is removed from So.
If no at 334, at 336 it is determined if the functionality is the exploration phase (i.e., whether 301 of
If yes at 336, at 340 a price p is chosen at random from the set of allowed prices for the item i.
If no at 336, at 338, from the set of allowed prices for the item i the price p is found that maximizes the objective value. In one embodiment, since there is usually a small number of price levels for each item i, all price levels for item i are tried in the objective function disclosed above, and the one that maximizes the objective function is chosen.
At 342, a new price vector, vnew, is found by setting the price of item i to p.
At 344, the value of the objective function fnew is determined at vnew.
At 346, it is determined if vnew is feasible and if fnew≧β*fcurrent. This decision accounts for the random choice for the exploration phase and because of inter-item constraints for both phases. The objective function algorithm checks only for business constraints feasibility, and not for inter-item prices. The inter-item constraints are accounted for at 332 where the set of allowed prices is constructed.
If no at 346, at 354 p is removed from Sa and the functionality continues at 334.
If yes at 346, at 350 fcurrent=fnew.
At 356, variable “n” equals the number of items with prices that differ from the original values.
At 358, it is determined if n is greater than a predetermined maximum number. If yes at 358, at 360, from the set of items with altered prices, the best item (excluding item i) is chosen that can be set to its initial price value. The “best” item is the one which results in the largest value of the objective function once this item is reset back to the price level in vcurrent.
At 362, it is determined if the best item is found. If no at 362, at 365 the changes done to vnew are revoked by resetting the price of item i to its initial value and then at 366 removing the current item i from So. If yes at 362, at 364 a new price vector, vnew, is found by setting the price of the best item to its initial value. Functionality then continues at 366.
If no at 358, at 366 item i is removed from So.
At 368, it is determined if So is empty. If no at 368, functionality continues at 330. If yes at 368, functionality continues at 316 of
At 316 of
As disclosed, one embodiment optimizes prices using two sequential phases, an exploration and exploitation phase, that alternate until there is a convergence to some locally optimal solution or until a maximum run time is reached. Each phase consists of repetitive cycles where the items are considered in random order using uniform probability distribution. In the exploitation phase, new solutions are accepted only if they improve the objective function. In the exploration phase, new solutions are accepted if they do not decrease the previously found best objective value by more than a specified percentage number.
The following pseudo-code can be used to implement the overview functionality in accordance with one embodiment:
The following pseudo-code can be used to implement the exploitation phase functionality in accordance with one embodiment:
The following pseudo-code can be used to implement the exploration phase functionality in accordance with one embodiment:
Embodiments can be applied to a large class of problems in addition to product pricing. Embodiments are especially suited for solving problems: that have a large number of input variable dimensions (typically hundreds or thousands), each of which can be either discrete or can be reduced to discrete; that are non-linear and non-convex; that include complex constraints among the function's input and output variables; and where there is no need for the optimal solution but for an improvement over the existing one.
An example of problems that can be ideally solved by embodiments of the present invention include a price optimization problem that includes thousands of dimensions (e.g., items in a grocery store). The item prices are constrained by “magic” numbers (e.g., $3.99, $4.99, not $4.31). The goal is to improve profit without changing initial prices of too many items. The need is to find a solution for a single store in 30 seconds.
Another example is a seating problem for a large corporate meeting. The problem includes thousands of dimensions (e.g., employees with different location possibilities). The goal is to improve the proximity of members within a working group. Constraints include only wanting to consider 100 moves or fewer. Another example problem is the allocation/reallocation of virtual machines in a cloud for improved resource utilization and energy savings.
Several embodiments are specifically illustrated and/or described herein. However, it will be appreciated that modifications and variations of the disclosed embodiments are covered by the above teachings and within the purview of the appended claims without departing from the spirit and intended scope of the invention.
