A Bayesian Network-based customer satisfaction model: A tool for management decisions in railway transport

Chakraborty, Subrata; Mengersen, Kerrie; Fidge, Colin; Ma, Lin; Lassen, David Article A Bayesian Network-based customer satisfaction model: A tool for management decisions in railway transport Decision Analytics Provided in Cooperation with: Springer Nature Suggested Citation: Chakraborty, Subrata; Mengersen, Kerrie; Fidge, Colin; Ma, Lin; Lassen, David (2016) : A Bayesian Network-based customer satisfaction model: A tool for management decisions in railway transport, Decision Analytics, ISSN 2193-8636, Springer, Heidelberg, Vol. 3, Iss. 1, pp. 1-24, https://doi.org/10.1186/s40165-016-0021-2 This Version is available at: https://hdl.handle.net/10419/185051 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Models ofsequential decision making inconsumer lending Kanshukan Rajaratnam1,2*† , Peter A. Beling1,2† and George A. Overstreet3† Background In consumer lending, portfolio managers typically have access to a scorecard used to forecast default probability for each applicant. Scorecards are built using historical data on loan accounts and their respective performance data. The inputs into the scorecard include financial, demographic and other personal information about each applicant.The output of the scorecard is a real-valued score for each applicant, which can then be mapped to a probability of default [see Hand and Henley (1997) for scorecard construction]. Similarly, portfolio managers may have access to a scorecard forecasting applicant responses to offers. In turn, forecasts of default probabilities and response probabilities serve as inputs to business metric functions such as expected profit. In setting loan prices (or loan interest rates), loan portfolio managers face a trade-off between response and risk. Consumers prefer lower loan rates and hence lower loan rates results in higher take-up of the products, but lower profits for each account. Loan pricing is further complicated by the phenomenon of adverse selection in which the default rates of individuals who accept a loan offer may be higher than that of those who decline the offer, all other factors being equal (Phillips and Raffard 2009). Adverse selection is thought to be the result of information asymmetry. Credit bureau reports and public records, which lenders use as input for credit risk and response models, may not reflect the circumstances and immediate financial needs of the borrower. Additionally, Abstract In this paper, we introduce models of sequential decision making in consumer lending. From the definition of adverse selection in static lending models, we show that homogenous borrowers take-up offers at different instances of time when faced with a sequence of loan offers. We postulate that bounded rationality and diverse decision heuristics used by consumers drive the decisions they make about credit offers. Under that postulate, we show how observation of early decisions in a sequence can be informative about later decisions and can, when coupled with a type of adverse selection, also inform credit risk. We show through two examples how lenders may use such information in setting their offer rates. Keywords: Sequential decision making, Consumer loans, Credit risk Open Access © The Author(s) 2016. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. RESEARCH Rajaratnam et al. Decis. Anal. (2016) 3:6 DOI 10.1186/s40165-016-0023-0 *Correspondence: kanshukan.rajaratnam@uct. ac.za †Kanshukan Rajaratnam, Peter A. Beling and George A. Overstreet contributed equally to this work 1 Department of Finance and Tax, University of Cape Town, Private Bag X3, Rondebosch 7701, South Africa Full list of author information is available at the end of the article Page 2 of 16 Rajaratnam et al. Decis. Anal. (2016) 3:6 there may be subtle relationships between price elasticity and adverse selection (Oliver and Thaker 2013). A portfolio manager may view a subset of the consumer population as homogenous as a result of the observable information available to him. However, private information held by individuals in the population subset differentiate their risk profiles. Portfolio managers do not have access to such private information. It would then seem obvious that reducing the information asymmetry between borrowers and lenders will result in more targeted marketing of credit products with appropriate rates. Typically, public information used by portfolio managers are those that are input into scorecards, i.e., financial, demographic and other personal information. The flat maximum effect indicates that, given roughly the same inputs, there is little difference in the performance of scorecards constructed using a variety of modeling approaches (Overstreet etal. 1992), i.e., new data sources or new variables need to be found in order to improve scorecard performance. An important line of research in human decision making is bounded rationality. Bounded rationality describe how a decision is made rather than the outcome of that decision (Selten and Gigerenzer 2002,p.4). It is the idea that decision makers are limited by the available information, time, cognitive ability, and the manageability of the problem. Humans use heuristics to make decisions, which are simple rules, but often lead to decision errors. Kahneman and Tversky were among the first to establish cognitive basis for errors arising from decision heuristics [see Tversky and Kahneman (1973) and Tversky and Kahneman (1974)]. Limited cognitive ability and incomplete information are some of the reasons for errors in decision making by human subjects. Many experiments have been conducted to reveal the decision heuristics used by human subjects in various classes of decision problems [seeWinkler and Murphy (1973) for more]. Given the diversity in the decision heuristics used by human subjects, this forms a new source of data that may be used to improve scorecard performance. One particular class of decision problems is the sequential decision problem in which agents are required to make a sequence of binary decisions. Sequential decision problems are of particular interest to consumer lending because consumers are often faced with a sequence of loan offers for which they make take/no take decisions. In this paper, we postulate that inference about the decision heuristics used by consumers when accepting or rejecting a loan offer may provide a new source of information for lenders. For example, decision heuristics could provide added information on borrower take/notake behavior, thereby reducing the information asymmetry between lenders and borrowers. In particular, we show how observation of early decisions in a sequence can be informative about later decisions and can, when coupled with a type of adverse selection, also inform credit risk. The paper is organized as follows. “Adverse selection” section extends the definition of adverse selection from Oliver and Thaker (2013) to a sequential offer setting that will serve as the basis for further study of the borrower’s decision process. “Bounded rationality” section discusses bounded rationality in human decision making and reviews literature on categorizing agents by their sequential decision making behavior. “Problems involving sequential decisions in lending” section introduces two sequential decision problems in the consumer lending space. The first decision problem relates to auction Page 3 of 16 Rajaratnam et al. Decis. Anal. (2016) 3:6 mechanisms for peer-to-peer lending. Lenders cognizant of decision heuristics in the context of consumer lending may offer a lower bid rate and, hence, win the bidding process. We derive policy implications for a marketplace desirous of increasing borrowers’ utility through lower interest rates. The second decision problem, set in the context of direct mail, is that of a lender required to choose when to market offers relative to the competition. We show how the lender may incorporate information learned about the decision heuristics of individual consumers. “Conclusion” section offers concluding remarks and suggestions for further research. Adverse selection In this section, we introduce basic notations, followed by the mathematical definition of adverse selection. We then extend notions of adverse selection to timing adverse selection (TAS) and provide motivation for the study of consumers’ decision making process in the consumer lending space. Suppose a portfolio manager has access to a homogenous population to which a credit product is marketed. We say a population is homogenous when members of the population have no observable differences between them. We use vector x to denote past behavioral, financial and demographic information about each member of this population and only denotes information observable by the lender. We call x , the characteristics vector. The portfolio manager makes an offer of credit with rate r. Once an offer is made, some subset of the population, the Take population, will accept the offer and open an account. Let T denote the event that an individual takes up an offer; so Tc is the event the individual declines the offer. Suppose this is a simple loan account, where a unit of loan is lent to each account holder, and the account holder is required to repay the unit of loan plus the interest on the loan, 1+r , at the end of a specified time period. We assume there are only two mutually exclusive and exhaustive performance outcomes, G and B. The event G is associated with a Good customer, being one who does not default within the specified time period and repays the loan in full. The event B denotes the performance of a Bad customer, being one who is not Good. We denote the probability of default for a borrower with characteristic vector and offer rate r as p(B|x,r) . The conditional probability of default for the Take population is then written as p(B|T,x,r) .1 Oliver and Thaker (2013) define adverse selection as Equation1 states the probability of a member in the Take population defaulting is higher than the probability of default in the general population, i.e., both the Take and Non- Take population. We use the total probability theorem to obtain the Bads among the Non-Takes, i.e., 1 At each instance of an offer, a homogenous subset of the total population (i.e., conditioned on the characteristic vector) offered the same rate (i.e., one rate at each instance of offer) will have some who take up the offer and some who do not. This take-up of an offer (or not taking up an offer) may be a result of unobservable information (i.e., unobservable to the portfolio manager). One may think of the random variable T as a function of some unobservable information provided by the take action that is independent of the characteristic vector and rate. (1) p(B|T,x,r)>p(B|x,r). (2) p(B|x,r)=p(B|T,x,r)p(T|x,r)+p  B|T c ,x,r  p  T c |x,r . Page 4 of 16 Rajaratnam et al. Decis. Anal. (2016) 3:6 Following (Oliver and Thaker 2013), Bayes’ rule can relate the conditional probability of Bad of a Take to the conditional probability of Take by a Bad, i.e., Since, p(T|x,r)+p(Tc|x,r)=1 , combining Eqs. 1 and 2 results in the following inequality: Equation4 indicates the Non-Take population have a higher credit quality than the total population. In defining Eq.4, we assumed the portfolio manager makes a one-time offer of credit. Suppose instead of a one-time offer strategy, the portfolio manager markets repeatedly. At each of a finite number of epochs, the manager has the option to market to individuals who have not previously taken an offer. Below we show that, due to adverse selection, the credit quality of those not-taking up any prior offers improve after every marketing instance. From Eqs.1 and 3, it follows that, where Ti is the random variable indicating take-up at the ith offer and ri is the offer-rate in the ith marketing instance. It follows from Eqs.4 and 5, that the credit quality of the non-take population after the first marketing instance is higher than the credit quality of the population prior to the first marketing instance, i.e., p(B|Tc 1,x,r1)<p(B|x,r1) , where Tc i indicates the event a borrower declines the ith offer. Suppose the portfolio manager markets a second time to those who did not take up the offer in the first marketing instance. It follows from Eqs.1 and 3 that, where ri is a vector of all past and current offers, i.e., ri={r1,r2,...,ri} . Equations4 and 6 can both be generalized for the ith marketing instance, i.e., and Note that Eqs.7 and 8 are extensions of Eqs.1 and 3. Equation7 implies that due to adverse selection, the credit quality of successive non-take population improves after each marketing instance. This is due to the higher probability of Take among Bads than (3) p(B|T,x,r) p(B|x,r) = p(T|B,x,r) p(T|x,r). (4) p(B|x,r)>p  B|T c ,x,r . (5) p(B|T 1 ,x,r 1 ) p(B | x,r1) = p(T 1 |B,x,r 1 ) p(T1 | x,r1) > 1, (6) p  B|T2,T c 1,x,r2  p  B|Tc 1 ,x,r 2 =p  T2|B,T c 1,x,r2  p  T 2 |Tc 1 ,x,r 2 > 1, (7) p  B|T c 1 ,...,T c i−1 ,x,r i >p  B|T c 1 ,...,T c i−1 ,T c i ,x,r i (8) p  B|Ti,T c 1,...,T c i−1,x,ri  p  B|Tc 1 ,...,Tc i−1 ,x,r i =p  Ti|B,T c 1,...,T c i−1,x,ri  p  T i |Tc 1 ,...,Tc i−1 ,x,r i > 1. Page 5 of 16 Rajaratnam et al. Decis. Anal. (2016) 3:6 the general population at each marketing instance. Furthermore, Eq.8 indicates there is a time component to adverse selection. We call this time dependent characteristic of adverse selection, timing adverse selection (or TAS). Note, thus far the vector of offer rates in ri has not been specified. Suppose some members of a marketed population decline all prior offers, it follows from Eq.8, It follows since p(Ti|Tc 1,...,Tc i−1,x,ri)≥0 , the probability of take-up for Bads is strictly positive and is greater than the probability of take-up among the general population. In such a scenario where some members of a marketed population decline all prior offers, there is a positive probability at each marketing instance of a Bad declining all prior offers and taking up the latest offer, i.e., Thus far, we have shown that given sequential offers, there is a positive probability at each marketing instance of Bads declining all prior offers and taking up the latest offer. This is derived from notions of adverse selection (see Eq.1) as discussed in Oliver and Thaker (2013). However, Eq.2 may be shown through an alternative classifier such as logistic regression built using the characteristic vector, past offers and performance data from a dataset built from an experiment. In order to create this data set, a portfolio manager requires access to a homogenous population. Given a homogenous population with characteristic vector, x , a portfolio manager may make a sequence of offers, where each offer in the sequence is only made to the subset of the population who decline all previous offers. Using this data, the portfolio manager may estimate the probability of Take and Bad given past declines, i.e., p ( Ti∩B|Tc 1, ... ,Tc i−1,x,ri) through a logistic regression scorecards.2 If p ( Ti ∩ B | Tc 1, ... ,Tc i−1,x,ri )> 0 , it follows that p ( Ti | B,Tc 1, ... ,Tc i−1,x,ri )> 0 (i.e., Eq.2). Equation is in line with observations of real-life subjects faced with sequential decision making problems in the context of consumer lending. Observations have shown homogenous subjects taking up offers at different instances of a sequence. In the next section, we take a borrower’s view of receiving a sequence of offers and discuss the decision heuristic observed in similar sequential decision making problems. Bounded rationality In “Adverse selection” section, we considered the case of a portfolio manager repeatedly marketing a credit product to a non-take population. The non-take population is updated after every offer is made. We showed under Oliver and Thaker’s definition of adverse selection [seeOliver and Thaker (2013)], the credit quality of the non-take population improves monotonically with marketing instance and that there is a timing aspect to adverse selection. (9) p  T i |B,T c 1 ,...,T c i−1 ,x,r i >p  T i |T c 1 ,...,T c i−1 ,x,r i. (10) p  T i |B,T c 1 ,...,T c i−1 ,x,r i >0∀i . 2 In order to estimate this probability, the portfolio manager requires all information regarding every offer made to each potential borrower. This requires each potential borrower to reveal every offer made by all institutions. The portfolio manager may make credit cards offers to a set of potential borrowers through direct mail channel, while incentivizing these potential borrowers to reveal offers from competitive institutions, whether these competitors offers were taken up or not. Credit performance of taken-up offers may be observed in credit bureau data. Page 6 of 16 Rajaratnam et al. Decis. Anal. (2016) 3:6 Suppose now we take a borrower’s view. A borrower receives a sequence of offers from a lender. We assume the offer expires before the next offer arrives. As each offer arrives, the borrower is required to make a decision on whether to take the offer. If an offer not taken, the borrower waits for the next offer. When rejecting an offer, the borrower risks the chance of receiving only lower quality offers in the future. A broad set of literature indicates that humans do not necessarily make rational decisions when faced with sequential decision problems, in part because of our bounded ability to take in information and limited cognitive abilities. Such limitations are known as bounded rationality. Bounded rationality may explain why homogenous borrowers accept credit offers at different point of time when faced with a sequence of offers. We provide an example, in the form of a well-studied problem known as the secretary problem, of how human subjects make decisions in a sequential decision problem setting. Decision heuristics The secretary problem, also known as the dowry problem, is a well-studied sequential decision problem involving optimal stopping theory. The secretary problem in its simplest form is as follows (Ferguson 1989). Suppose a manager wishes to fill a secretarial position. There is only one such position available, for which there are N applicants. The manager is aware of the number of applicants. We assume the applicants can be rankordered from best to worst candidates without ties. The applicants are then interviewed sequentially and in a random fashion. Once an applicant is interviewed, the manager is required to make a decision to hire the applicant or not. If the applicant is hired, no further interviews takes place. However if the applicant is not hired, the decision maker interviews the next candidate. Rejected applicants cannot be recalled. The objective of the manager is to hire the best possible applicant. After each interview, the manager faces a trade-off, i.e., the manager could hire the current interviewee and risk the chance that a better applicant would have arrived later on in the interview process, or not hire the current interviewee but no higher quality applicant arrives later. The optimal solution can be described using the idea of a candidate. An applicant is a candidate if he or she is the best applicant interviewed thus far. The optimal solution is then for the manager to reject the first h−1 applicants, some integer h≥1 , and then choose the next candidate (Ferguson 1989). Let N denote the number of applicants. For N>1 , the probability of selecting the best applicant is, The optimal solution is h∗=argmaxtφN(t) . This is easily solved for small values of N. As N→∞ , h∗=N/e (Ferguson 1989). It follows that for large values of N, it is approximately optimal for the manager to interview 36.8% of the applicants and then select the φ N(h)= N  j=h p(jth applicant is the best applicant and is selected ) = N  j=h1 Nh−1 j−1 =h−1 NN  j = h 1 j−1. Page 7 of 16 Rajaratnam et al. Decis. Anal. (2016) 3:6 next applicant better than all previously interviewed applicants. The probability of successfully choosing the best candidate is approximately 36.8% [see Ferguson (1989), and Gilbert and Mosteller (2006) for more]. Stewart (1981) extended the secretary problem to one where the number of options is unknown. Under the assumption arrival times of each option is independent and identically distributed exponential random variable, the probability of choosing the best candidate with such a policy is 1/e, which is the asymptotic optimal probability value for when the length is known (Stewart 1981). Because of bounded rationality and behavioral biases, humans do not necessarily make decisions in a rational manner. Experiments in decision making with real-life subjects have shown diverse decision making heuristics. A field experiment by Seale and Rapoport (1997) is particularly important because it demonstrates that when people were presented with the secretary problem, they did not generally behave optimally but rather in fashions that could be explained as mixtures of three decision heuristics, each with a parameter. The decision making strategies reported by Seale and Rapoport (1997) are: 1. Cutoff rule Reject the first h−1 applicants and then hire the next candidate. 2. Successive non-candidate rule Hire the first candidate who follows h successive noncandidate applicants since the last candidate. 3. Candidate counting rule Hire the hth candidate. Note that of the three decision rules, only the cutoff rule is optimal, and then only if the correct parameter is chosen. Seale and Rapoport (1997) observed that human subjects seemed to follow a mixture of rules, with mixture weights and parameter values varying across individuals. We speculate that multiple decision heuristics are in use by individuals responding to sequential credit offers. Such decision heuristics found among borrowers might explain why timing adverse selection occurs in practice. Furthermore, values for heuristics parameters might correlate with notions of patience on the part of the borrow, an idea explored below. Credit hunger In a field experiment with low to moderate income households, Meier and Sprenger (2010) tested whether time preferences can explain credit behavior. They measured time preferences of individuals through choice experiments. The choice experiment outcomes were then matched to credit report and tax return data. After controlling for disposable income and other characteristics, less patient individuals were found to have lower credit scores and higher default rates. While Meier and Sprenger’s field experiment did not control for credit score, we posit that even when individuals do not have any observable differences, impatient consumer behaviors lead to higher default risk. We call this credit hunger. If, as we speculate, credit hunger exists in consumer credit populations, there would be value in recognizing individuals with that characteristic. Methods for learning decision strategies from the observation of actions could provide such an ability. In the next section, we introduce recent work in machine learning that addresses related problems. Page 8 of 16 Rajaratnam et al. Decis. Anal. (2016) 3:6 Behavior‑based agent recognition Suppose there exists a set of decision heuristics, similar to those observed by Seale and Rapoport (1997), governing consumers’ decision making processes. Any lender that could gain the ability identify the decision heuristics being used by individual borrowers might then be able to achieve an advantage in lending strategy, relative to competitors without that ability. Recent work in machine learning has addressed a class of problems called behavior-based agent recognition (BAR), which center on the recognition of decision strategies (or the identity of agents) based on observation of decisions made by agents in sequential problems. Qiao and Beling (2013) address the BAR problem by modeling the decision problem faced by agents as a Markov decision process (MDP). They use inverse reinforcement learning (IRL) to the learn the reward vector of the MDP from the observed actions of the agents. The reward vector is, in turn, used as the feature space for supervised and unsupervised learning of decision agent identities. On several problems, feature spaces constructed from rewards learned from IRL outperform those constructed directly from observed actions (Qiao and Beling 2013). For the secretary problem, Qiao and Beling (2013) conduct a simulation experiment in which a distinct base parameter value was applied to each heuristics rule from Seale and Rapoport (1997). In addition, random noise was added to actions of the decision agents. The feature space learned from IRL resulted in clusters with high-accuracy relative to ground truth. The method did not require inputs on any description of the decision heuristics as a basis for recognition. Suppose historical data of consumers’ accept/reject decisions for a sequence of offers was available, including related historical account performance. In such a scenario, using Qiao and Beling’s IRL model-based method, it might be possible to cluster consumers based on their decision heuristics. In addition, using the historical account performance, one could relate risk and response behavior to individual decision heuristics as well as the historical proportion of the borrower population using each decision heuristics. Furthermore, in identifying decision heuristics of historical population, distribution of parameter values for each decision heuristics could be estimated. A portfolio manager with access to such information might then incorporate his knowledge of the borrowers’ decision heuristics in the consumer loan offer strategy. In the next section, we discuss the impact of decision heuristics and parameter value information on a lender’s decision. Problems involving sequential decisions inlending In this section, we introduce two sequential decision problems found in consumer loan settings. In the first problem, we introduce the lending process in a social lending platform, where lenders offer loans to borrowers through a bidding process. We model the offer policies of portfolio managers cognizant of notions of credit hunger and the resulting impact on the final-rate offered to the borrower. Whereas in the first problem, lenders were merely cognizant of credit hunger, in the second problem we assume lenders have access to greater information such as the distribution of decision heuristics and the distribution of heuristics parameter values found in a borrower population. The portfolio manager is required to decide whether to market a credit product to a homogenous Page 15 of 16 Rajaratnam et al. Decis. Anal. (2016) 3:6 this phenomenon through bounded rationality resulting in diverse decision heuristics used by consumers. Along with decision heuristics in consumer lending space, we introduced the notion of credit hunger. This was followed by introducing a method used to cluster agents based on their decision heuristics—behavior-based agent recognition. Finally, the paper illustrated the impact of credit hunger and decision heuristics on two decision problems in the consumer lending space. As with adverse selection, timing adverse selection is not easy to measure in practice. However, sequential decision making experiments in other settings provide evidence that humans employ diverse decision heuristics, and this in turn suggests the existence of timing adverse selection. In order to categorize historical borrowers into clusters of decision heuristics, both take and non-take decision information is required. While offers taken-up by borrowers are found in credit bureau records, to our knowledge lenders do not share information on past declined offers. In addition, determining consumer lending decision heuristics requires an audit of offers, take behavior for all offers, and account performance for every offer accepted by a consumer. While we have shown examples of the impact of timing adverse selection on a lender’s decision, field experiments and further research is required in order to understand the phenomena described in this paper. In testing timing adverse selection and in determining consumers’ decision heuristics, a sequence of offers need to made and consumers’ decisions recorded. Such a sequence of offers may be disrupted by other lenders marketing their own products. The set of potential borrowers must then be incentivized to reveal all competitors’ offers whether they are taken up or not. Direct mail channels for credit cards lends itself to such an experiment. Authors’ contributions All authors contributed equally. The idea was generated by Peter Beling. The work was written by all three co-authors. All authors read and approved the final manuscript. Author details 1 Department of Finance and Tax, University of Cape Town, Private Bag X3, Rondebosch 7701, South Africa. 2 Department of Systems and Information Engineering, University of Virginia, 151 Engineer’s Way, Charlottesville, VA 22904, USA. 3 McIntire School of Commerce, University of Virginia, Charlottesville, VA 22904, USA. Acknowledgements This work is based on the research supported in part by the National Research Foundation (NRF) of South Africa for the Grant No. 93649. Any opinion, finding and conclusion or recommendation expressed in this material is that of the author(s) and the NRF does not accept any liability in this regard. Competing interests The authors declare that they have no competing interests. Received: 20 October 2015 Accepted: 6 November 2016 References Beling P, Overstreet G, Rajaratnam K. Estimation error in regulatory capital requirements: theoretical implications for consumer bank profitability. J Oper Res Soci. 2010;61(3):381–492. 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