Decision Optimization

BACKGROUND OF THE INVENTION

TECHNICAL FIELD

The invention relates generally to optimization techniques. More particularly, the invention relates to a computer implemented method and apparatus for providing an optimal solution for a large consumer decision.

DESCRIPTION OF THE PRIOR ART

A large consumer decision is characterized by a decision made for millions of accounts with one or more global constraints. The inclusion of the global constraints means that the decisions for all accounts must be examined all together. Such problem in academics has been labeled an NP-hard problem (it is formulated as a 0,1 -linear program), because the combinations exponentially rise based on the number of accounts and the number of decision alternatives for each account. A typical solution to the problem can be described with reference to Fig. 1, a schematic block diagram showing components of a solver for a consumer decision according to the prior art. Samples or segments of account 2 data are provided as input into a calculator 4 which consists of user defined scoring functions and system functions. The output of the calculator is used to construct the objective, constraints of the 0,1- linear program of the consumer decision model. The relaxation of the linear program is the input to a linear program (LP) solver 8. The LP solver outputs a solution which might contains fractions 10. It should be appreciated that some custom solvers enforce a limitation on the number of variables or type of constraints allowed. This problem has been around for the last ten years, i.e. since the consumer- marketing sector started to look at using optimization algorithms to assist with decisions on what to offer consumers. The credit card industry, for example, has been a leader in applying these algorithms.

The current approach in the industry uses sampling and segmentation techniques when applying optimization algorithms to large scale consumer decisions that include global constraints. In sampling techniques, the solution depends on the quality of the samples. In segmenting techniques, the solution depends on the quality of the segments. Such sampling and segmentation techniques are used due to limitations of existing optimization algorithms and computer memory. Most algorithms require the entire problem to be loaded into memory. These decision problems do not fit into a 32 bit address space. On occasion custom code has been written that may reduce the problem in other ways, such as only requiring one global constraint and narrowing the solution to solve only one specific type of decision. Another approach includes using 64 bit computers. However, such computers are expensive and the solvers are either new or inexperienced.

It should be appreciated that all such approaches achieve varying degrees of success, depending on the experience of the people applying the approaches and the applicability of the sampling, segmentation, and/or problem reduction techniques. None are fully acceptable for today's projects.

Y. Galperin, V. Fishman, and L. Gibiansky, Method for Optimizing Net Present Value of a Cross-Selling Marketing Campaign, WO0111522 (published Feb. 15, 2001) discuss an iterative algorithm to the problem of multidimensional optimization of cross-selling. The techniques discussed therein describe a solution by supplying a non-linear mathematical formulation, the non-linearity being due to introducing the Lagrange multipliers, to the traditional linear multidimensional problem desired to be solved when offering a large number of promotions M to a very large set of prospective customers N. Such process consists of randomly selecting a statistically significant sample of a prospect list, calculating the value of a utility function for each pair of an offer and selected prospects, reducing the original linear multidimensional problem to a smaller problem (still linear) with a feasible number of dimensions, solving the smaller problem for the selected sample numerically with the desired tolerance using an iterative algorithm, and using the results to calculate a set of offers in one pass for the full prospect list. It should be appreciated that Galperin, et al only use a sample of data, not all of the data, and do not guarantee the optimality, just a solution.

It would be advantageous to solve very large optimization problems at the account level.

It would be advantageous to provide an algorithm that takes advantage of the structure for the consumer decisions and the criteria used to evaluate the decisions, and, such that while taking all data into consideration, the algorithm formulates a much smaller problem to feed a solver.

It would be advantageous to provide a method and apparatus where the size of the problem fed to a solver can be configured.

It would be further advantageous to provide an algorithm that is able to find the global solution to the problem initially posed, even with the smaller problem fed to the solver.

It would be advantageous to provide a solver that removes the dependencies on third parties.

SUMMARY OF THE INVENTION

An iterative approach to solving the optimization problem is provided. Each iteration solves a relatively small sub-problem compared to the entire problem size. Yet, the algorithm still finds the global solution to the problem initially posed, thereby allowing better approximation to the overall large consumer decision problem. In a consumer decision model, a client/customer is represented by a row record. A partition of row records of the entire problem into disjoint sub-sets is called a row partition. An element in a row partition is called a segment. The invention provides an iteration of four basic operations; determining a row partition, balancing the row partition, expanding selected segments to create a new row partition, and solving the newly created row partition. The method and apparatus can use any off-the-shelf linear programming solver, such as Dash Optimization Xpress by Dash Optimization, during the solve operation. The size of the problem fed into the solver remains bounded and can be reduced to where heavyweight solvers like Xpress and ILOG CPLEX by ILOG, Inc. are not required, thus removing dependency of third parties. It should be appreciated that such solvers add special heuristics to address large complex problems. The size of the problem fed into the solver remains bounded and relatively small compared to the entire problem size. Thus, the algorithm can solve problems of several orders of magnitude larger. This ability to solve larger problems enables existing sampling and segmentation approaches to use more samples and segments, thereby providing a more precise solution. In one embodiment of the invention, the sampling and segmentation techniques are removed to where the problem is solved at the account-level. In the above cases, the solution is produced in a more cost-effective manner and the best possible return is achieved because the doubt of achieving a true global solution is removed if the solver called within the algorithm returns an optimal solution.

BRIEF DESCRIPTION OF THE DRAWINGS

Fig. 1 is a schematic block diagram showing components of a solver for a consumer decision according to the prior art;

Fig. 2 is a schematic block diagram showing the solver for a large consumer decision according to the invention;

Fig. 3 is a flow diagram of the scalable exact algorithm according to the invention; and

Fig. 4 is a detailed flow diagram of the scalable exact algorithm according to the invention. DETAILED DESCRIPTION OF THE INVENTION

An iterative approach to solving the optimization problem is provided. Each iteration solves a relatively small sub-problem compared to the entire problem size. Yet, the algorithm still finds the global solution to the problem initially posed, thereby allowing better approximation to the overall large consumer decision problem. In a consumer decision model, a client/customer is represented by a row record. A partition of row records of the entire problem into disjoint sub-sets is called a row partition. An element in a row partition is called a segment. The invention provides an iteration of four basic operations; determining a row partition, balancing the row partition, expanding selected segments to create a new row partition, and solving the newly created row partition. The method and apparatus can use any off-the-shelf linear programming solver, such as Dash Optimization Xpress by Dash Optimization, during the solve operation. The size of the problem fed into the solver remains bounded and can be reduced to where heavyweight solvers like Xpress and ILOG CPLEX by ILOG, Inc. are not required, thus removing dependency of third parties. It should be appreciated that such solvers add special heuristics to address large complex problems. The size of the problem fed into the solver remains bounded and relatively small compared to the entire problem size. Thus, the algorithm can solve problems of several orders of magnitude larger. This ability to solve larger problems enables existing sampling and segmentation approaches to use more samples and segments, thereby providing a more precise solution. In one embodiment of the invention, the sampling and segmentation techniques are removed to where the problem is solved at the account-level. In the above cases, the solution is produced in a more cost-effective manner and the best possible return is achieved because the doubt of achieving a true global solution is removed if the solver called within the algorithm returns an optimal solution.

Commercial Advantage

It should be appreciated that a commercial advantage is provided that allows an enterprise and its customers to solve existing problems with more precision, thereby providing the customers with a greater return. If an enterprise currently employs sampling or segmentation techniques, then one embodiment of the invention allows such projects to use many more samples or segments, possibly an order of magnitude of more samples or segments. Furthermore, an enterprise and its customers can begin to address projects that require account-level decisions with global constraints, such as cross-sell. Current techniques cannot adequately address such projects due to the type of local constraints required, which forces true account level optimization. Current techniques being applied to cross-sell use sampling, segmentation, approximations, or a reduction in ability, such as using only one global constraint, such as volume.

Overview

One embodiment of the invention can be described with reference to Fig. 2, a block schematic diagram showing the solver 12 for a large consumer decision according to the invention.

One embodiment of the invention provides the following features:

• It wraps existing solvers;

• It can be used on all processing levels: o account; o sample; and o segment.

• It uses four main operations in an iterative manner: o finding a row partition; o balancing the row partition; o expanding the row partition to create a new row partition; and o solving the newly created row partition. A high level flow ot one embodiment of the invention can be described witn reference to Fig. 3, a flow diagram of the scalable exact algorithm 100. The algorithm begins by computing an initial solution of the input problem (102). Set the initial solution as the current solution. Then the algorithm proceeds with the following: (1) Determines, or finds, a row partition based on the current solution (104). (2) After finding a row partition, balances the row partition (106). In the balanced row partition, marks all non-singleton segments as non-expanded or "usable". (3) Determines in among the segments weather there are any usable segments (108). If not, then the algorithm ends (110). (4) If there are usable segments, then selects a limited number (preconfigured) of such segments and expands them to singletons (112). A new row partition is produced. (5) Applies a solver to solve the newly created row partition (114). Then, given the solved row partition, the algorithm determines if a better objective function value or better solution is obtained (116). If yes, the algorithm uses the returned solution from the solver to update the current solution, and goes to (1). If not, control goes to (3).

An Exemplary Scalable Algorithm

The algorithm proposed here is for the account level formulation and can be applied to sample-weighted and segment formulations. For simplicity, we assume that there is no exclusion of treatments from any account. If not the case, it can be taken care of in the implementation of the algorithm.

The new algorithm takes an iterative approach. In each iteration, it maintains a row partition M and a basic solution B of OLP[M] such that the "fractional," defined hereinbelow, segments of M regarding to B are singletons. Then the algorithm tries to refine M through solving a smaller linear program that is constructed based on a row partition. The new algorithm reduces the memory usage by the third party linear program solver dramatically. Generally speaking, the memory usage of a third party solver will not be a bottleneck for the new algorithm. It opens the door to using less powerful but robust other third party linear solvers.

In the next several subsections, some necessary notions and a few atom operations or subroutines are introduced that act as the building blocks of the main algorithm. Account level linear program formulation.

Denote the linear relaxation of the account level formulation as the original linear program (OLP) that is stated here:

OLP:

maximize^ PyXy y Subject to

where I is the index set of accounts, T the index set of treatments, and G the index set of global constraints.

Reformulate the OLP by adding a penalty variable to each global constraint as follows:

maximize J] Pjjxy - J] psys ij s Subject to

Xy ≥O for Vz G /, Y/ eT and ys ≥0,se G.

where each ps is an extreme large positive number acting as the penalty weight to a non-satisfied global constraint. OLP can still be used to denote this formulation. Both formulations are equivalent if the first formulation has a solution. But the second formulation has the advantage that it always has a solution. I he algorithm stated in this document uses the second formulation as its input format.

The linear program from a row partition.

Let MAXSIZE be a positive constant integer that represents the maximum size of a segment in a row partition. Set it as 1000 for now.

Let M= {Mi,M2, ---M$ be a row partition of the entire input row records (or input accounts). Let | Mq | denote the cardinality of Mq or the frequency of Mq. If a segment contains only one element, then it is called a singleton. Again let T be the

index set of possible treatments. Let « = 1 / I M I ∑ pif and qj q ieMg 1J

ws j = 1/ 1 M I ∑ af- where /?•• and af- are the same as in OLP. ιeMq Lex ULr[Mj αenoie me ronowing linear program, a segment rormuiation oaseα on ιvι:

OLP[M]:

maximize

Subject to

y^ > 0 for q = l,2,...,k and V/ ≡ T,ys ≥ 0,s e G.

Let B be a solution of OLP[M]. It is called a basic solution if the components of B can be partitioned into two parts: basic and non-basic. The non-basic components are zeros, for the problems herein, and the columns of the coefficient matrix of OPL[M] corresponding to the basic components, or basic variables, forms a maximal independent set (as independent vectors). A basic solution exists (refer to Vasek Chvata. Linear Programming. W. H. Freeman and Company, 1980). Such concept is critical to the algorithm for obtaining an optimal solution. Let B be a basic solution of OLP[M]. For a segment Mq in M, let bvidex(Mq )={j\ ify^ is a basic variable in B}. It is called the basic treatment index set of Mq. If bvidex(Mq ) has more than one element, then Mq is called a "basic fractionaf (in short, "fractionaf) segment regarding to B. The "re-balancing" operation on M defined hereinbelow is for these non-fractional segments, i.e. their basic treatment index set containing only one element. This operation makes the size of a non-fractional segments in check. Let fd] denote the smallest integer that is greater than or equal to d where d is a real number.

Re-balancing(M,B): Repeat { if there is a non-fractional segment Mq of M whose size is greater than MAXSIZE, split it into f | Mq|/MAXSIZEi segments as equal as possible; set tne Dasic treatment inαex set or eacn new segment to oe Dvιaex(Mq j. Delete A/q from M. Add the newly created segments to M. } until no segment meeting the "if" condition". Repeat { if the sum of their sizes of two non-fractional segments of M is less than or equal to MAXSIZE and their basic treatment index sets are the same, then merge them into a new segment. Set the basic treatment index set of the new segment to be the basic treatment index set of the two selected segments. Delete the two selected segments from and add the new one to M. y until no more segments meeting the "if" condition. Return M.

Note that all the balancing operations are performed on these non-fractional segments. After re-balancing, a new row partition is produced. For this newly created row partition, the following routine computes a basic solution of OLP[Re- balancing(M,B)] from B.

Basis-Balancing(Re-balancing(M,B),B): Let B={xiJtyε}, M={MlfM2,...,Mk} and Re-ba\anc\nq{M,B)={W1,W2,..., Wn}. Set yqj =(\ if there is / such that WqcMi, otherwise YoJ = σ X-- for q=l,2,...,n and je T.

Return { yqj,ys} .

The following operation/subroutine expands a row partition M to a new one by replacing a segment Mq of M with the individual records it contains.

Expanding(M, Mq): Remove Mq from M and add each element of Mq as a segment to M. Return M.

Let B = (VqJ1 ys } be a basic feasible solution of OLP[M] and Mq is a non-fractional

segment regarding to B. Let S(M, N\q) be the set of singletons in Expanding(M, Mq).

Let M* denote the set of these non-singleton segments of Expanding(M, Mq). Define

OLP[M, MφB] to be the linear program deduced from OLP[M] and OLP as follows: OLP[M, Mq,B]: maximize ∑ ∑ P^, . - ∑ psys + ∑ ∑ u y

Subject to ft) ∑zhi =l> VheS(M,Mq), j J ftϋ σ σ aL zhi -ys=bs - ∑ ∑ ^77-V S&G> zh . ≥O for \/h&S(M,Mg) and \/j&T,ys >0,seG.

It should be appreciated that OLP[M, Mq,B] is defined on S(M, Mq). It can be viewed intuitively that OLP[M, Mq, B] is obtained by projecting OLP[M] onto S(M,Mq). It has a feasible solution (if let zhj= (1/|Mq|)Vq/ for h eMq and zhi =vhJ if heS(M,Mq) and h £Mq , a feasible solution is obtained and it has the same objective value as B).

Let W=expanding(M, Mq) and B', a basic solution of OLP[M1Mq1B]- Let Basis(W, OLP[M1 Mq ,B]) denote {vhj|h in M*}uB\ It can be shown that Basis(W, OLP[M, Mq ,B]) is a basic solution of OLP[W]. The notation is used in the main algorithm, described hereinbelow.

Row partition based on a basic solution.

If M is the trivial row partition, i.e. each segment contains one account, then by the same notation used in the previous subsection, OLP=OLP[M]. From now on the account-level formulation can be viewed as nothing else but a special case of the row partition or segment formulation.

Let M be a row partition. Let B be a basic solution of OLP[M]. For each segment, it has a basic treatment index set as defined in the previous subsection based on B. Every non-basic component of B is zero. Now the following operation or subroutine that generates a row partition based on B can be stated as follows:

Segments(M,B): (1) Expand all fractional segments of M regarding to B to singletons; (2) Partition the non-fractional segments of M into subsets such that any two segments belong to the same subset if their basic treatment index sets are the same (containing only one element). Merge tne segments in eacn suoset to rorm a new segment. (3) Return the segments newly created in (1) and (2).

It should be appreciated that the merger operation is only performed on the non- fractional segments regarding to B. Based on this fact, the following subroutine computes a basic solution of OLP[Segments(M,B)]:

Basis_Micro(M,B): Let B={x13,ys} be a basic solution of OLP[M]. Let Segments(M,B)={Wi,W2,...,Wk} and M={M1,M2,...,Mn} For each segment Wq, if Wq is also a segment of M with index i, set yqj= xu otherwise set yφ= ∑ x . M^Wq

Return B*={ yq],ys}.

Disaggregating a segment level solution.

Let M be a row partition and B={VιPγs } an optimal solution of OLP[M]. Again let / be the index set of accounts and T the index set of treatments. Let P be the index set of segments of M. Below is an example sub-routine to disaggregate B to a solution of OLP in the following:

DisAggregating(M,B): Compute the map f from I to P such that f(i) =q if / belongs the segment Mq indexed by q- Compute U(βJ ={ulk~ vf(,)k/\Mf0)\ \i e I, k e T}. Return U(B) plus {ys}.

Computing an initial basic solution of OLP.

In this subsection, a method to compute a basic solution of OLP is proposed.

InitialBase(OLP): Generate a row partition M (can use a Mont Carlo method); boive ULPLNJ ;

Let B be the returned basic solution of OLP[M]. Let W be the row partion computed as follows: For each segment Q in M { If Q is non-fractional regarding to B { Add Q to W; } else { Add each element of Q as a singleton segment to W (i.e. expanding Q); }

In OLP[W], for each non-singleton segment Q, fix its treatment assignments to the same value as in B (this means to introduce more constraints). Solve OLP[W] with the additional constraints. Let B* be the returned basic solution. Return DisAggregating(W,B*);

In an example prototype of the algorithm, the very first row partition is generated in the following way:

Without considering the global constraints, assign the best treatment for each row record or account. Then partition the row records or accounts according to their treatments assigned such that two row records are in the same subset if they have the same treatment assigned. Then balance the row partition to produce a desirable row partition.

The size of the linear programs in the subroutine can be controlled by the maximum size of a segment along with the fact that there are only a few fractional segments regarding to a basic solution. This means that the linear program OLP[W] is not that big and can be solved efficiently by a third party linear program solver.

The algorithm. The algorithm starts with a basic solution of OLP, solves, by calling a third party linear program solver, a sequence of linear programs and arrives at a solution of OLP. If the third party solver returns an optimal and basic solution in each call, then the algorithm will return an optimal solution to OLP. As a by-product, it also finds an optimal row partition, i.e. the optimal solution of the linear program based on this row partition can be converted or disaggregated to an optimal solution of OLP.

The algorithm can be described with reference to Fig. 4, a workflow diagram for the algorithm, along with the corresponding pseudo-code presented hereinbelow. The algorithm is referred to herein as Zheng Account-Segment-Generation Algorithm (ZASGA). ZASGA works for the soft constraint setting, i.e. the penalty weights may be set to be relatively smaller in considering the trade-of between the global constraints and the business objective, as well. For initial input, OLP is computed by, for example, Fair Isaac Corp.'s Decision Optimizer, using account level information 402. The output of the algorithm is a row partition and a solution of OLP 412. The returned solution is optimal or not totally dependent on the solver ZASGA calls as indicated above.

ZASGA pseudo-code:

Input: OLP computed by using account level information 402. Output: A row partition and a solution of OLP 412. Data Object Definition: B: A double array to store a basic solution. B*: A double array to store a basic solution. M,W: Vector of sets to store segmentations. Obj(B): a double to store the objective value computed using a basic solution B. BestObj: a double, the best objective value computed so far, initialized to the minus infinity. Mark: Boolean array with the same length as M to represent the "usable" and "unusable" segments information. MAXSIZE: a constant integer representing the maximum size of a segment. NUMITERATIONS: set to be a large integer, it is the limit of possible iterations of the major loop

Algorithm Body: t3<HnitialBase(ULP) 4U4. W <- the trivial row partition- each segment is a singleton 404; iternum<-0406; While ( iternum < NUMITERATIONS) { // begin-while itemum<-itemmu+1 ; If (BestObj<Obj(B) 418) { Bestθbj<- the objective value of B; M<-Segment(W,B) 406 B*<-Basis_Micro(W,B) 406; M<-Re-balancing(M,B*) 406; B*<-Basis-balancing(M,B*) 406; For (each segment i ) { if it is not single-ton, Mark[i]=true 408; //usable else Mark[i]=false 408; //unusable } } If (there is a usable segment) 410 { Select a usable segment Mq from M 414; Mark[q]=false 414; W<^Expanding(M, Mq) 414; Compute OLP[M, Mq 1B*] 416; Solve OLP[M, Mq ,B*] 416; Compute B=Basis(W, OLP[M1 Mq ,B*]) using the returned basic solution of OLP[M, Mq ,B*] and B* 416; } else 410{ Return M and disAggregating(M,B*) 412; } } // end-while 418

It should be appreciated that the method assumes that the input linear program is not degenerated. In the case if the input linear program is degenerated and the solver called within the method returns a basic optimal solution, in order to guarantee an optimal solution ot tne original linear program, one more oooκ-κeepιng step neeαs to be added: i.e. after last 416, disaggregating the basic solution B=Basis(W, OLP[M, Mq, B*] to obtain a basic solution of the original linear program, say Bλ. If Bλ is different from the current best solution, but its objective value may be the same as that of the current best solution, then store, i.e. keep, W and B. Then before exiting the method, check if there are stored W and B. If yes, replace the current solution and row partition by the stored B and W and go to 406. Otherwise exit the method.

To reference detailed proofs of the steps of the algorithm hereinabove, refer to the internal paper of Fair Isaac Corporation to Maolin Zheng, A Scalable Exact Algorithm For Decision Optimization (Revised Version), July 2005, which paper is incorporated herein by this reference thereto.

It should be appreciated that the operation/subroutine Re-balancing(M) keeps all intermediate linear programs in the algorithm to a predictable size. Such is very important for the scalability of the algorithm. Also in the algorithm, only one non- singleton is selected to expand in each iteration. Actually several such segments can be selected to expand at the same time in order to reduce the number of total iterations if there is enough computer memory permitting to do so.

It should be appreciated that if the solver called in operation "Solve OLP[M, Mq ,B*]" 416 returns a basic optimal solution of the sub-linear program OLP[M, Mq ,B*], then the algorithm returns an optimal solution of OLP, the original linear program.

Solving mixed integer programs.

In one embodiment of the invention, a large consumer decision problem is formulated as 0,1 -linear program (if restricting all the decision variables in OLP as 0,1 ) and propose an algorithm to solve its linear relaxation.

Take M and B=disAggregating(M,B*) from the return of ZASGA, if B contains no fractions, then B is an integer solution to the large consumer decision problem. Otherwise, all the fractional segments are singletons. It can be proven that there are at most |G| such fractional numbers in B where )G| is the number of global constraints. To reference the detailed proofs, refer to the internal paper oτ ι-aιr isaac Corporation to Maolin Zheng, A Scalable Exact Algorithm For Decision Optimization (Revised Version), July 2005, which paper is incorporated herein by this reference thereto. For example, if there are 2 million accounts and 4 global constraints, the result is at most 4 accounts with fractional assignments. By rounding these fractions to their nearest integer, an integer solution is obtained. It might be not optimal or feasible. But the error is relatively smaller comparing with the scale of the problem. Another way of obtaining an integer solution is to solve OLP[M] by calling an MIP solver of a third party and disaggregate the solution to an account level solution. Because OLP[M] is relatively small, the MIP solver shall solve it. The disaggregated solution might not be optimal but shall be very good. The quality of the solution, optimal or not, is dependent on the solver called in the algorithm.

Example Results

Table A shows the results of using the computer environment: 1.2 GHz CPU, 1.2 GB RAM, Windows 2000 and applying 12 treatments and 4 global constraints for all runs.

Table A

Table B shows the results of using the computer environment: 1.2 GHz CPU, 1.2 GB RAM, Windows 2000 and applying 12 treatments and 4 global constraints for all runs.

Table B

It is evident from the discussion and results shown hereinabove that benefits to an enterprise using the method and process provided herein include more samples translate to better precision for projects, for example better forecasts, increased constraint satisfaction, and increased client satisfaction and conversion rates. More segments translate to better differentiation of treatments, for example better forecasts, reduced model and prediction error, and increased client satisfaction and strategy control.

It should be appreciated that in one embodiment of the invention performs the operations in a parallel manner. It is possible to split the processing of an inner loop into threads that independently run the expand and solve operations. Another embodiment of the invention uses alternative selection algorithms for finding the usable segment in which to expand.

Another embodiment of the invention uses alternative algorithms for seeding the basic feasible solution.

Another embodiment of the invention tunes the cache size used to feed the LP solver.

Accordingly, although the invention has been described in detail with reference to particular preferred embodiments, persons possessing ordinary skill in the art to which this invention pertains will appreciate that various modifications and enhancements may be made without departing from the spirit and scope of the claims that follow.