Introduction

Retail banking has evolved over time from being transaction-centric to a relationship-centric business. Banks now see the merit in nurturing and growing profitable long-term relationships with their customers. The field of customer relationship management (CRM) has grown considerably in recent years specifically to deal with this challenge.

The above statement regarding profitable relationships presupposes the notion of the value of a relationship, and a metric that allows the business to quantify this value. Once such a metric is defined, the objective of the business becomes one of maximizing this value over its customer base. The most intuitive definition of relationship value is one of profit, since this is in line with the business goal of profit maximization.

There are various factors that influence the profit potential of a customer. From a revenue standpoint, the driver is the availability of the right product at the right time at the right price in order to maximize both the propensity to take the product and to use it profitably. From a default risk standpoint, it is the ability to determine the right level of exposure for the bank (quantified in terms of credit limits), and the right price (quantified in terms of a risk premium over the prime lending rate). From an attrition standpoint, it is the ability to gauge the level of profitability of the relationship for the customer, and choose the right CRM action in order to make it beneficial for both sides to continue.

At a higher level of abstraction, the challenge is to optimize in the presence of resource constraints such as the availability of capital, the portfolio-level risk preference, regulatory requirements for economic capital allocation and the budget allocation for CRM actions. These constraints imply the necessity of choosing between customers on the basis of the factors discussed above. Thus, the business may 'choose to lose' certain customers whom it does not deem sufficiently profitable in the long term, while concentrating its efforts and resources on retaining the customers who are most likely to have a sustainable and profitable relationship with it.

Figure 1 depicts the central problem of customer relationship management described above. The objective, as summarized in this chart, is to find the right product to sell to the right customer at the right price at the right time.

Figure 1.

Towards customer relationship value maximization.

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A fairly common application of this concept is the process of cross-selling. For instance, customers who hold sales finance loans with a bank are encouraged to take personal loans/credit cards as well. The term 'cross-sell' is often used to refer to the case where a customer holding one product (say, a credit card) is sold a different product (say, a personal loan). The term 'up-sell' is used when the customer is sold more of the same product. We shall use the term 'cross-sell' in a loose sense here, wherein a customer who holds one product with the bank is sold another product, which may or may not be of the same type that he already holds. The primary advantages of an effective cross-sell process are as follows:

Retention. For potentially profitable customers who currently hold closed-end products such as loans, it gives the business a way of retaining these customers and sustaining the relationship.

Profit maximization. It is often the case that many customers in a consumer finance business are acquired through not-so-profitable products offered during festive seasons in order to swell volumes. In order to make these relationships profitable in the long run, one would need to find a good product with high profit potential to cross-sell to the customer.

While this activity can be done on a continuous-time basis, it is often organized in the form of regular campaigns, such as during the festive season, in order to take advantage of the spending habits of customers during those times.

The process followed during a typical campaign can be described as follows: The bank has a list of customers from which it can choose to contact a subset with an offer for a new loan. A common way of contacting the customer is to send him a letter indicating that he is being offered a particular product, say a personal loan. The letter may also indicate a pre-approved limit, that is, the maximum amount of the loan that the customer can take. Other methods of contacting the customer, such as telemarketing, can also be used.

The bank needs to select from among the list of customers it has on its books, the ones that it would like to target during a particular campaign. In order to do this, the bank requires a way to make a distinction between the customers in its database, on some characteristics that are relevant to the decision problem at hand. The process of selecting customers for a cross-sell campaign therefore comprises two main steps:

Estimation. The first step is to analyse the customer's information available with the bank and predict certain characteristics, such as the propensity of the customer to respond favourably to a cross-sell offer, his profitability given response, etc. Various statistical and data mining techniques may be used to get estimates of these quantities. Several published works talk about these prediction techniques (Rygielski et al, 2002).

Target selection. Based on these estimates, the bank uses some method to come up with the list of customers to whom the cross-sell offers are to be communicated. The bank may communicate one or more product offers at a time to a customer. This is essentially an optimization problem, involving some business objectives and constraints, as well as customer-specific inputs relating to response propensity, profit potential, risk, etc (obtained from the previous step).

In spite of CRM being quite popular in the retail banking industry, the published work related to mathematical methods for target selection is scarce. There are some papers that discusses similar problems in other areas such as the direct mail retail industry (Kalvaitis and Posgay, 1974; Dwyer and Evans, 1981; Gonul and Shi, 1998), or papers that discuss CRM in general (Lewis, 2005; Akcura and Srinivasan, 2005). Campbell et al (2001) and Storey and Cohen (2002) have formulated target selection as mixed integer programming problem. Storey and Cohen (2002) have also discussed the computational issues that arise due to the large number of decision variables involved, a typical phenomenon in the retail banking industry. However, to the extent of our knowledge, the existing literature in this area has not dealt with the fact that the customer or segment-level parameter estimates that go into the optimization model as inputs are usually not very accurate. The unreliability of these estimates in turn affects the reliability of the results given by the optimization model.

In this paper, we propose a mathematical programming method that takes care of the computational complexity of the problem as well as the unreliable parameter estimates.¹ We address the issue of computational complexity issue by providing a group-level formulation instead of a customer-level formulation, and the issue of unreliable estimates by using fuzzy numbers to represent both these estimates and the constraints applied on them.

The use of fuzzy numbers leads us to represent the problem as one of fuzzy mathematical programming. There have been works related to the properties and solution methodologies for solving fuzzy linear programming. Zimmermann (1976, 1978) proposed a method for linear programming problems. Similar methods were proposed in Rommelfanger (1996), Gasimov and Yenilmez (2002) and Fullér and Zimmermann (1993). We formulate the problem as a fuzzy integer linear program. The fuzzy problem is then transformed to a multi-objective problem with crisp numbers. The multiple objectives deal with the most likely of the fuzzy numbers as well as their spread in either direction. This multi-objective problem is then solved using a method proposed by Young-Jou and Ching-Lai (1994) that attempts to maximize the worst case satisfaction level of all the objectives.

The organization of this paper is as follows: We describe the cross-sell problem in detail in the next section. Subsequently, we provide two mathematical programming formulations of the problem, both of which assume that the estimates of the customer-level parameters are crisp numbers. We discuss the issues with these formulations, and address them by restating the problem as one of fuzzy mathematical programming. We then demonstrate the utility of the method presented in this paper through an example.

Problem description

This paper discusses the cross-sell problem of a bank with a set of customers on its books, whom it would like to target with more product offerings. Let us assume that more than one product may be offered to each customer. The choice of product offered to a given customer can be said to depend on three factors:

1. The response propensity of the customer to a given product.
2. The profit potential, given response to that product.
3. The risk of default (and the loss given default) on that product, given response.

One may also consider additional criteria such as the volume of credit extended to the customer; in this paper, we consider the problem of cross-selling closed end products, that is, loans, hence the additional criterion boils down to the expected volume of the cross-sold loan given response. If the profit metric being used is risk-adjusted, that is, if it accounts for profit as well as loss, then the risk factor may be omitted in the decision making. The problem being discussed in this paper assumes the use of such a profit metric.

It is clear that the bank would like to target customers who are both likely to respond and likely to be profitable. Therefore, the decision problem involves an understanding of the trade-off between response propensity and profit potential, should there be a case where the two are not concordant. Additionally, in the case of a choice between multiple product offerings, the trade-off between these factors for different products is also to be understood.

As discussed in the previous section, the key parameters (response propensity, profit potential, etc) are predicted based on the past information available about customers. On the input side, this information can be account level, demographic or behaviourial information about the customers. On the output side, this would be response to past offers, and the profit derived thereafter. Various methods can be used to find a relationship between these inputs and outputs, so that these parameters can be predicted for the customers who are candidates for cross-selling.

Based on these predictions the bank would like to decide whether to contact a customer with the offer or not. The costs associated with each offer are the following:

Targeting cost (TargCost). This is the cost of contacting the customer through one or more channels—direct mailing, telemarketing, etc.

Operating expenses (OpEx). This is the cost of opening an account, and servicing the new cross-sold product.

A third kind of cost may be the risk which the bank would be subject to, in case the customer defaults or prepays the loan. However, this may be implicitly indicated in the profit measure, as earlier specified.

We consider the final objective of the bank to be to maximize its net income (NI) from a cross-sell campaign. Intuitively, we understand the term 'net income' to mean the algebraic sum of inflows and outflows. The specific inflows and outflows being considered, however, depend on the business context. For instance, one bank may chose to allocate overhead such as employee salary and origination cost at a per account basis, whereas another bank may not do that. In the context of the problem considered in this paper, the inflows to the NI calculation come from the profit realized from the converted loans, whereas the outflows comprise the targeting cost and operating cost (overhead such as origination cost and employee salary). A bank may chose to optimize the NI after tax; however, we are not considering this as the tax component in a particular business is constant and just a multiple to the NI figure. We represent the NI as given in Equation (1):

where E(Profit) is the expected profit from the campaign. It is clear that the expected profit figure from a campaign depends on both response propensity and profit potential, the operating expense depends on the expected response, and the targeting cost depends on the number of customers targeted.

Certain other business constraints may also have to be met while making the final selection of customers for a campaign. We consider two major constraints in this paper, namely the campaign budget and the business' asset growth target. It is assumed that there is a fixed cost for contacting a customer;² therefore, the campaign budget translates to an upper bound on the number of customers who can be targeted. Secondly, the bank would like to increase its net earning assets (ie, total loaned volume) through the campaign. Therefore, the expected volume booked by all the targeted customers should meet the asset growth target. The second constraint implies that we need estimates of loan volume given response for each customer, in addition to the profit and response estimates. This entire process, including the estimation step, is described in Figure 2. This paper concentrates on the 'target selection' part of the problem, where we assume that we have the estimates of response propensity, profit scores and the expected volume of each customer in consideration.

Figure 2.

Cross-sell process.

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In the subsequent sections, we present some mathematical programming formulations to solve the above problem. We discuss the advantages as well as the issues associated with these formulations and demonstrate them using an example.

Crisp formulations

The problem described above may be modelled as a constrained mathematical programming problem. The objective is to maximize the NI of the bank from a campaign. There are J types of products on offer in the campaign and N customers to be considered. We assume that we have the estimates of the three important parameters (response, profit, volume) for these customers. We first present a formulation at the customer level; however, while this would be the ideal level of granularity for decision making, it has certain issues regarding the reliability of estimates and computational complexity, hence we also present a more realistic approach that makes decisions on groups of customers.

Both approaches discussed in this section regard the parameters being used as crisp numbers.³ We assume that response propensity is represented as a probability value; note, however, that there may be other methods of arriving at customer-level estimates wherein the outputs are not crisp numbers. For instance, using Bayesian belief networks for estimation might give us profit and volume distributions for each customer, using classification and regression trees might give us interval-valued estimates, and so on (Breiman et al, 1984; Jordan, 1998; Li et al, 2004). In this paper, we concentrate on formulations that use point-valued (crisp) estimates of profit and volume scores, such as those obtained using regression techniques.

Customer-level formulation

An ideal way of solving this problem is to take a decision for each customer. The decision to be made is the offer to be given to a customer, including the case where no offer is made. So we define the decision variable y_ij as follows:

Let r_ij be the profit score for customer i for product j. Similarly p_ij and v_ij are the estimates for response probability and volume, respectively. _j represents the operating cost and m represents the targeting cost per customer. Based on this, the basic formulation of the problem would be as follows:

The objective can be written as maximize the NI, which can be stated as follows:

subject to the following constraints:

There are two major issues with this formulation. The first issue is the scale of the problem. This is a 0/1 mathematical programming problem with the number of decision variables equal to the product of number of customers in the database and number of products to be offered (N J). If N is large enough, the scale of the problem may pose computational difficulties. The second issue is the reliability of the inputs to this model. As discussed in the previous section, the inputs are the estimated values of some key parameters. It is very difficult to get accurate estimates for these parameters at the customer level. However, we observe that, if we group the customers into quantiles on any given parameter, the quantile-level aggregate estimates perform better in terms of the averages and the rank ordering. Since the quality of the solution given by the optimization model is dependent upon the quality of inputs, it is more meaningful to formulate the problem at a group level rather than at a customer level. So, the main difference between a customer-level and a group-level formulation is that the group-level formulation provides the decision for a group at a time compared to a decision for each customer given by a customer-level formulation.

Group-level formulation

As discussed, a more realistic approach is to formulate the problem at a group level. Since the bank would like to select customers with a good combination of profit score and response probability, we rank the customers based on each parameter and group those with similar values.⁴ For instance, if we consider two products and divide the customers into deciles on profit score and response probability for each product, the total number of groups would be 10 10 10 10 which is 10 000 groups. Therefore, we now need to make decisions for these 10 000 groups. We define a decision variable x_ij as the number of offers of product j given to a group i.

In some cases, we can offer more than one product to a customer. This somewhat complicates the situation as the targeting cost for sending offers for multiple products is the same as that for one product, so capturing this information at a group level is difficult. This situation can be easily handled by introducing the combination of products as new products and changing the formulation of the asset growth constraints accordingly.

The average of the response rates of customers in a group can be taken as the representative response rate for the group. However, we take the average of p_kj r_kj, kGroup i as a representative figure for the expected profit of group i, since the product of the averages is not equal to the average of the products. A similar calculation applies to the expected volume to be used in the asset growth constraint as well. Let P_ij, R_ij and V_ij denote the average response rate in group i for product j, the average expected profit and the average expected volume, respectively.

Let N_i denote the size of group i. The group-level formulation of the problem is as follows:

The objective is to maximize the NI from all the groups (z^g)

subject to the following constraints:

The solution of this model would give us the number of offers of each product to be given to each group. If x_ij=N_i, all the customers in group i are offered product j. However, if 0<x_ij<N_i, then we solve another problem of distributing the offers (x_ijj) to the customers in the group. This is an assignment problem that can be solved easily.

There are a few issues associated with the formulation and the input parameters that will be discussed in the next subsection.

Issues with the group-level formulation

The inputs to the optimization problem are the predicted values of response propensity, profit score and volume score. These are inherently difficult quantities to predict for a customer; hence, the predicted values of the parameters are usually not very reliable. Since these values are used as inputs to the optimization problem, the results of the optimization step would in turn not be very reliable. The first step towards addressing this issue is to group customers and use the group averages as the inputs, thereby trading precision for accuracy. However, the group-level formulation given in the previous section uses imprecise inputs without a change in the solution methodology. The issue of imprecision is not directly addressed in target selection. For instance, consider two candidate solutions, one with a higher expected NI and higher imprecision associated with it, and another with lower expected NI and lower associated imprecision. The current method simply considers the expected NI and chooses the one with the higher quantity. This choice, however, does not take into account the risk appetite of the decision maker.

One method for capturing imprecision is to define the parameter values as fuzzy numbers. The advantage of using fuzzy numbers as the inputs is two fold. In problems involving estimates of parameters relating to human behaviour, it is more intuitive and acceptable if the prediction is given in terms of a fuzzy number. For instance, if we predict the profit potential of a customer as $x, it will be less acceptable as compared to a prediction that the profit potential of a customer would be around $x (represented as ). Secondly, the output of the fuzzy mathematical program gives an objective value which is a fuzzy number. This gives a more realistic estimate of the NI one can expect from the campaign, by providing an optimistic, pessimistic as well as the most likely scenario.

In the following section, we define the input parameters as fuzzy numbers and solve the associated fuzzy mathematical programming problem.

Fuzzy mathematical programming formulation

In the formulation given in earlier sections, the parameters in use are: profit score, response probability and volume. We consider the estimates of profit and volume as triangular fuzzy numbers. We also consider the asset growth targets for each product as triangular fuzzy numbers. As described earlier, we distribute the customers in different groups formed on the basis of profit potential and response probability values. We find out the average, minimum and maximum of each group. The support of the fuzzy numbers is defined by the minimum and maximum of profit and volume, respectively, for each group and the average of the group is represented by the most likely value (having membership value equal to 1). The fuzzy number for the targets is defined by taking some inputs from the decision maker.

We assume a triangular fuzzy number for the fuzzy estimates of the input parameters for the sake of simplicity and explanatory properties. However, other membership functions can be used as well. As defined in Zimmermann (1991), the membership functions allowed for a fuzzy number have to be convex with a unique point with membership value of 1. Any membership function which follow these criteria can be used for this problem. Linear membership functions have been discussed in Zimmermann (1976, 1978). An interval linear membership function has been used by Hannan (1981) and a piecewise linear membership function has been used by Hu and Fang (1999). Bhattacharya et al (2008) and Vasant et al (2007) have proposed logistic-based membership functions for a product-mix problem. The membership function can also be customized depending on the problem context (Bhattacharya and Vasant, 2007).

The fuzzy mathematical programming problem can be formulated as follows. The objective function can be written as

subject to the following constraints:

So, Equations (8), (9), (10) and (11) represents formulation of the fuzzy mathematical programming problem. As discussed earlier in this section, there might be some ambiguity in the estimates of the profit score and volume score. The profit score can be represented as a fuzzy number⁵ _ij=[R_ij^p,R_ij^m,R_ij^o], where

This is just one way of representing the fuziness in the estimates. This is the most intuitive way of representing the estimates as fuzzy numbers. One might use some other method based on the domain knowledge.

Similarly we define _ij=[V_ij^p,V_ij^m,V_ij^o] for expected volume. The asset growth target for product j is represented by _j=[T_j^p,T_j^m,T_j^o]. Therefore the constraints can be written as follows:

We then use a method similar to the ones proposed by Buckley (1988) and Lai and Hwang (1992) to convert this single objective fuzzy programming problem to multi-objective crisp programming problem.

Let us begin with the objective function. The objective function of the problem can be represented by a triangular fuzzy number as shown in Figure 3. The two extremes of the fuzzy number represent the optimistic (z^o) and pessimistic (z^p) values of the objective, while the mode (z^m) gives the most likely value. We wish to maximize this objective function. This can be achieved by a set of three crisp objectives: maximize the most likely NI (z^m), minimize the negative spread of the NI (z^m-z^p) and maximize the positive spread of the NI (z^o-z^m). Based on the business requirements, we can even restrict this to a two-objective problem. For example, if the business is concerned about maximizing the most likely NI and minimizing the downside risk and not really bothered about the upside gains, we can convert the fuzzy objective to a two-objective problem where the two objectives are: maximize z^m and minimize z^m-z^p. But in this paper we take the general case with three-objective problem. Therefore, the fuzzy objective functions boils down to optimizing the following three objectives.

Now we discuss how to take care of the fuzzy constraints. Since fuzziness is associated with the asset growth targets only (and not the budget), there are J fuzzy constraints in the above formulation. These are converted to crisp constraints using the method proposed by Tanaka et al (1984) as written in Equation (14). The other two sets of constraints (9) and (11) remain as they are because there is no fuziness involved in those constraints. So the entire set of constraints can be written as follows:

Thus, the single-objective fuzzy programming problem is now converted to a multi-objective crisp programming problem, with three crisp constraints for each fuzzy constraint. In the subsequent section, we discuss a method to solve this multi-objective problem and arrive at the final target list.

Figure 3.

Spread of the NI.

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Solving the multiple objective problem

We use the method proposed by Young-Jou and Ching-Lai (1994). We solve the problem considering one objective at a time.⁶ The corresponding optimal objective value is represented by f_i^*, and the values of the decision variables is represented by X_i for each objective i. The other objective functions are also calculated at X_i and are represented by f_j(X_i) where ji. We denote f_i' as the minimum (maximum) value in each column for the maximization (minimization) objective while solving the model for other objectives. These values are arranged in Table 1.

Table 1 - Multiple objectives.

Full table

Then the problem is reformulated as follows. The objective function is written as

where

The f_i^*-f_i' term in Equation (15) provides the maximum extent to which one will have to compromise on the ith objective; therefore, maximizing amounts to maximizing the extent to which one can avoid compromising on all objectives at the same time. Also, since f_i' represents the worst one can do on the ith objective and f_i^* represents the best one can do, represents the worst case satisfaction level across all the objectives, and the original problem becomes transformed to one of maximizing the level to which we are satisfied with achieving the multiple objectives. Another way of interpreting the above formulation is that _i provides the equivalent of the membership value of the ith objective, so one is trying to maximize the -cut on all three objectives at the same time.

This problem is solved along with the original constraints of the model. The advantage of this solution methodology over goal programming is that the model can be solved for simultaneous goals without assigning the subjective weights. At the same time, one can also assign subjective weights to the objectives by manipulating the _i formulae appropriately. One could do this by changing the definition of _i to

where _i represents the relative importance of satisfaction of the ith objective.

The entire solution methodology can be described as shown in Figure 4. This is a very high-level description of the method.

Figure 4.

Block diagram for the solution methodology.

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In the following section, we demonstrate the method described above by solving a real cross-sell problem for a bank.

An Example

In this section we demonstrate the method by solving a cross-sell problem of a bank. In the interest of maintaining confidentiality of the data, we have anonymized the products on offer and the currency of transactions. We solve the multi-objective problem using the method mentioned in the previous section and then analyse the results to get a better understanding of the solution.

Experimental setup

We have taken a small subset of the entire customer database of a bank to demonstrate this method. This is done to ensure the confidentiality of the data. The subset has 184 115 customers. The bank has two products on offer, which we shall refer to as A and B. One or more products can be offered to a customer. We include some aggregate level information of the data. For each customer, we have estimates of profit, volume and response propensity for each product. The quantile-wise averages (we consider the averages of ten percentiles or deciles) of response and profit scores for the two products are given in Figures 5, 6, 7, 8.

Figure 5.

Product A: Average response propensity by decile.

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Figure 6.

Product A: Average profit score by decile.

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Figure 7.

Product B: Average response propensity by decile.

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Figure 8.

Product B: Average profit score by decile.

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The budget constraint allows us to target a maximum of 150 000 customers. It should be noted that the targeting cost is incurred only once even if we target the customer with both products. Therefore, for the purpose of formulation, we consider it a problem with three products: A, B and A+B, with an additional constraint that a customer can be offered at most one of these products. Additionally, in order to account for cannibalization of one product by another when a joint offer is made, we discount the response propensity of each product in case of a joint offer by an empirically determined factor. The asset growth target for product A is around 7MM⁷ and for product B is around 4MM. The fuzzy numbers for these targets are defined as [5,7,10]MM for product A and for product B as [2,4,7]MM. We then follow the following steps to solve the problem:

1. Arrange the customers in decreasing order of profit and Response Probability. Divide them into 10 groups based on profit and Response Probability deciles. So all the customers are distributed in 10 000 groups. Note that all these groups may not be populated. Based on the data we have, only 9685 groups were populated. Therefore, we have 3 9685=29 055 decision variables.
2. For each group we calculate the minimum, maximum and average values of all the parameters. We also calculate the number of customers in each group. The fuzzy numbers for profit and volume are defined as triangular fuzzy numbers, with the support between minimum and maximum of each group.
3. We formulate the fuzzy mathematical program and then convert it to a multiple objective crisp programming problem. Now we have three objectives with 29 055 integer variables and 9692 constraints.
4. We solve the program for each objective independently.
5. We use the method explained in the previous section to reformulate the problem with one objective and solve the problem to get the final solution.

All the integer programming problems were solved using ILOG OPL Studio with CPLEX 10.0 solver. A Windows 2000 based machine of 3 GHz processor and 4 GB of RAM was used. The total time taken to solve the problem was around 75 s (18 s each for the three single objective problems in Equation (13) and 21 s in solving the last problem). After solving the final problem, the objective value achieved is [1.602,2.28,4.51] MM with total number of offers as 150 000. The satisfaction level () achieved is 0.714.

Table 2 gives a comparison of the results obtained from the crisp group-level formulation and the fuzzy formulation. The analysis of these results is given in the following subsection.

Table 2 - Comparison of results.

Full table

Analysis of results

We perform the analysis of these results with respect to three aspects of the solution: objective function values, constraint satisfaction and decision variables.

Objective function. If we compare the NI figures, we find that both the values are very similar. If we compare the most likely NI in the fuzzy formulation (z^m) to the NI in the crisp formulation (z^g), z^m is around 3.8% lesser than z^g. This is because the fuzzy formulation attempts not only to maximize the expected NI, but also the optimistic and pessimistic case. Under certain conditions, it may happen that some sacrifice is made in z^m (vis-à-vis z^g) in order to gain on z^p and/or z^o. To verify this, we take the decision made using the crisp formulation and calculate the result that this decision would have produced, had we used the fuzzy formulation instead. The fuzzy number obtained for NI for that decision is [1.58,2.382,4.831] MM. When we compare this result with that of the fuzzy formulation, we find that the fuzzy model does indeed compromise on z^m, but produces a better result for z^p, that is, the downside spread of the fuzzy solution is lesser. We also find that the vagueness or the overall spread in the NI has gone down from 3.25 MM in the crisp case to 2.908 MM in the fuzzy case, which is a reduction of around 12%. As discussed earlier, the decision maker may be interested in the spread as much as in the expected value, therefore, this method provides a benefit in that sense.

Also, when we analyse the support of from the fuzzy formulation, we find that the spread of the number in the negative direction is much lower compared to its spread in the positive direction. The support of a fuzzy number represents the vagueness associated with the number. So, a higher spread on the positive direction can be taken as the number being more likely to be on the positive side. So the fuzzy formulation is more likely to give a higher NI than z^m.

Constraint satisfaction. We observe that the optimistic case constraint on the asset growth of product B is tight in the fuzzy solution. Since product B is found to be less profitable than product A on average, this is the toughest constraint to meet: therefore, the optimizer ensures that it barely meets this constraint on product B, and the remaining allocation is biased towards the more profitable product A.

Decision variables. Now let us compare the number of offers given for each product. The number of offers for product A has gone down by 4.7% when we formulate the problem using fuzzy numbers. The number of offers for product B has gone up by 5.2%. This shows that product A seems to be more profitable when we consider the averages of the profit scores, but not as profitable when we consider the spread of these figures in a group as well. To have a better understanding of the method, we analyse the groups where the decisions given by the two methods are different. Out of the total of 9685 groups, 559 groups have different decisions. When we compare the spread in the profit scores in each group for product A, for these 559 groups, we find that the upside and downside spread are more or less even, whereas on average, the spread is skewed towards the upside for the entire population. Therefore, on these groups, the decision is different on account of the fuzzy formulation wanting to maximize the optimistic case NI as well.

Sensitivity analysis

We can perform sensitivity analysis to see the effect of change in various parameters on the final result. To demonstrate this, we have done sensitivity analysis on the budget and the pessimistic volume target for product A. Similar analysis can be done for the other parameters, based on the business needs.

Budget constraint. We now discuss the effect of changing budget constraint on the results. The results discussed above is based on a budget constraint of 150 000 (fund for contacting 150 000 customers). We change the budget from 140 000 to 180 000 with a change of 10 000 in each instance. The results are summarized in Table 3.

Table 3 - Sensitivity analysis: budget.

Full table

It is evident from the results that as we increase the budget for the campaign, the total number of offers as well as the NI keeps on increasing. If we concentrate on the last row of the table (with budget constraint as 180 000), the total number of offers given is only 171 751 (=102 187+21 052+48 512). This means that there are not enough profitable customers to use up the whole budget. We also note that Product B offers have gone down by around 2%. This is primarly because of the first objective of minimizing the downside spread. The downside spread of the NI for budget constraint of 170 000 is 0.69 MM (=2.33-1.64). The NI in the last row has gone up in total, but the downside spread of the NI remains at the same level. This is done by sacrificing a few offers of Product B on groups of customers having a higher downside spread, in favour of product A.

Volume constraint. We now discuss the effect of changing asset growth target on the final result. There are two asset growth targets in this problem: target for both the products. These targets are represented as triangular fuzzy numbers. For this demonstration we change the pessimistic asset growth target for product A and analyze its effect on the results. The pessimistic asset growth target for product A in the problem discussed above is 5 MM. We change this between 4.5 MM and 6.5 MM with a difference of 500 000 in each instance. The results for this study are summarized in Table 4.

Table 4 - Sensitivity analysis: volume.

Full table

It is clear from the results that the number of offers for product A keeps on increasing to satisfy the increasing volume constraint for the product. But since the number of offers to be given remains the same (150 000 offers, sum of the three types of offers), the higher number of offers for A is compensated by lower number of offers for product B. The number of offers for B is kept at a level where it satisfies the asset-growth constraint.

Similar sensitivity analyses can be performed to serve different business needs and to support the decision maker in making efficient and effective decisions.

Conclusions

In this paper we have discussed the target selection process for a cross-sell campaign using unreliable estimates of customer-level parameters. We have formulated the problem at a group level, thereby eliminating some computational complexity and using more accurate, albeit imprecise, estimates for the input parameters. To take care of the imprecision, we have represented these parameters as triangular fuzzy numbers and solved the resultant fuzzy constrained programming problem.

The bank would like to maximize its NI from a campaign; however, at the same time, it may also desire minimum ambiguity in these figures, especially on the downside. The fuzzy formulation takes care of this objective by maximizing the pessimistic NI estimate as well. In order to do this, it may, however, need to compromise on the mean NI objective. While the method as applied in this paper gives equal importance to all three objectives (maximizing the optimistic, pessimistic and most likely case), it is flexible enough to incorporate information about the risk appetite of the decision maker by changing the _i formulae appropriately (see Equation (16)).

A few directions for future research are proposed here. At the formulation level, one may consider options wherein the spread is directly optimized, instead of simply maximizing the entire fuzzy number . For instance, the three objectives used could be max z^m, min z^m-z_p, max z^o-z^m.

One of the key components of the solution is the method in which customers are grouped. The current method allows us to form homogenous buckets based on the objective of the campaign; however, it may lead to a large number of groups when the number of products (J) increases, thereby leading to computational difficulties. Better grouping schemes may have an impact on the quality of the solution.

Another aspect is the construction of fuzzy numbers based on the range of parameter values within a group. While this range captures some of the imprecision, it does not adequately capture the error in estimating the input parameters. For instance, the range of profit scores in a bucket may be between 80 and 120, whereas the range of the 'true' profit values is likely to be between 60 and 150—by using methods such as resampling, one may be able to improve the specification of the fuzzy numbers used in the formulation to account for this difference.

At a more fundamental level, one may look at other ways of capturing the uncertainty in the parameters. For instance, one may use prediction methods such as Bayesian belief networks that output profit scores in terms of probability distributions, and then use stochastic programming techniques to solve the problem. The computational complexity of these techniques has always been a challenge. One can try evolutionary computation or other computational intelligence techniques to take care of this issue.

Notes

¹ Discussion on methods for estimation of the parameters is outside the scope of this paper.

² There can be different channels for contacting the customer. Finding the optimal channel could also be part of the problem, but that is outside the scope of this paper.

³ We refer to these as crisp formulations. Later in this paper, we propose a fuzzy formulation, where we assume some inputs to be fuzzy numbers.

⁴ One may also consider other grouping strategies. For instance, since the objective of the optimization problem is to maximize NI, one may group customers based on their NI on various products. However, given that decisions can be made on subsets of a group rather than for the group as a whole, it does not seem that a different grouping strategy will materially affect the solution.

⁵ We represent triangular fuzzy numbers using the notation [a,b,c], where a represents the lowest point of support, b represents the mode and c represents the highest point of support.

⁶ Most of the OR techniques used to solve multiple objective problems try to optimize one objective at a time. However, there exist methods that solve multiple objectives at the same time (eg, Genetic Algorithms for solving multi-objective optimization problems; Deb et al, 2002).

⁷ MM=million.

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