From share of choice to buyers' welfare maximization: Bridging the gap through distributionally robust optimization

Abstract

We study the product design problem where a decision maker selects the features of a product from a set of feasible options. We focus on two widely studied objectives in this field, that is, the share of choice (SOC) and buyers' welfare (BW). The two objectives are vulnerable to different types of customer preference misspecification, that is, deviation from the nominal utility distribution and effects of outliers, respectively. We formulate a distributionally robust optimization (DRO) SOC maximization model and a winsorized BW maximization model to obtain robust solutions to the two problems. Interestingly, we show that the two robust models are equivalent in a certain sense—for appropriate choices of robustness parameters, both models return the same solution. This observation has important ramifications. For instance, it indicates that a product designed to yield higher BW is more robust to the deviation from the nominal utility distribution for the SOC problem, while that with higher SOC is less sensitive to the effect of outliers for the BW problem. Last but not least, we use the equivalence to develop a new approach to solve the DRO model for the SOC problem, using a version of the winsorized BW model. Extensive numerical experiments demonstrate the superior performance of the proposed approach.

INTRODUCTION

Companies often rely on product strategy to gain prominence in selected markets (Xia et al., 2016). Failure to design a good product often causes firms to suffer direct financial losses and indirect damages to reputation (Wang & Curry, 2012). To select the “right” product, decision makers must clearly understand customer preferences that influence choices for alternative products. Conjoint analysis (CA) ¹ is a common technique often used to elicit customer preference (measured by utility values) from a representative sample of the population. Since the publication of Zufryden (1977), extensive prescriptive models based on CA have been established in both operation research and the marketing field (Bertsimas & Mišić, 2016; Green & Krieger, 1985; Jiao & Zhang, 2005; Kohli & Sukumar, 1990; Shi et al., 2001). Depending on the number of products, the models are divided into single‐product design, or product‐line design (multiple products are designed simultaneously) (Shi et al., 2001). Our paper first focuses on the single‐product design problem for ease of exposition, and provides extensions to the product‐line design problem in the Supporting Information (EC.1.2).

The share‐of‐choice (SOC) problem is one of the most widely studied problems in the literature (Camm et al., 2006; Kohli & Sukumar, 1990; Shi et al., 2001; Wang et al., 2009; Wang & Curry, 2012; Wang & Gutierrez, 2022). The problem aims to find a product that maximizes SOC, that is, the percentage of customers for whom the offered product delivers higher utility than the outside options, using the point estimation of individual partworth (utility values) attached to each feature (estimated from CA) for the representative sample of the population (Balakrishnan & Jacob, 1996; Balakrishnan et al., 2004; Kohli & Sukumar, 1990; Shi et al., 2001). The implicit assumption under the above approach is that the partworths are estimated accurately, and their distribution over the sample represents the entire population.

However, the partworths are often not measured precisely for two reasons. First, it is hard to get an accurate point estimation of partworth at the individual level, due to the limited range of questions in the CA survey (Wang & Gutierrez, 2022) and the limitations of estimation techniques (e.g., the hierarchical Bayesian method provides only a posterior distribution instead of a point estimator for the utility of each customer; Allenby & Rossi, 1998). On average, the hit rate of customer choice based on CA data is only 75%–85% (Johnson & Orme, 2010). Second, because only a subset of the entire population is included in the CA (Bertsimas & Mišić, 2016), the well‐known optimizer's curse (Smith & Winkler, 2006) holds if we optimize the objective over the sample distribution. This phenomenon has actually been observed in previous product design literature (Schön, 2010a).

Recently, several papers studying the SOC problem focus on customer preference uncertainty. Camm et al. (2006) and Wang et al. (2009) solved the classical model multiple times (via sampling) and use the majority vote to choose a robust solution. Wang and Curry (2012) and Wang and Gutierrez (2022) applied different robust frameworks to find robust solutions during the optimization phase. In the two studies, robustness is obtained by allowing the “true” partworth of each feature to vary in an “uncertainty set” around the nominal value, and the product with the best worst‐case performance is selected. All the literature showed the superior performance of the robust solutions, which supports the necessity of incorporating robustness against customer preference uncertainty in the SOC problem.

Camm et al. (2006), Wang et al. (2009), and Wang and Curry (2012) reported the detailed specification of the solutions with different extents of robustness. The data are publicly available, which allows us to evaluate the performance of the robust solutions. In Figure 1, we compare the buyer's welfare (BW) metric, which is another widely studied objective, among the solutions with different levels of robustness. BW refers to the summation of utility surplus generated due to customers choosing the offered product over the population (Balakrishnan & Jacob, 1996; Balakrishnan et al., 2004; Green & Krieger, 1985). For the solutions in these studies, we arrange them in ascending order of the extents of robustness (in Camm et al., 2006; Wang et al., 2009, the extent of robustness is indicated by the frequency of the solution to be optimal in the robustness test; in Wang & Curry, 2012, the extent of robustness is measured by the range of value that partworth can vary); that is, the solutions on the right‐hand side of each figure are supposedly more robust compared to those on the left‐hand side of the same figure. Surprisingly, in Figure 1, all the results indicate that a more robust solution usually attains higher BW. This raises the following question: Is the positive correlation between BW and the robustness of the SOC problem a fundamental phenomenon or merely a coincidence? We answer this fundamental question in the paper.

FIGURE 1

BW of solutions reported in Camm et al. (2006), Wang et al. (2009), and Wang and Curry (2012): (a) Camm et al. (2006), (b) Wang et al. (2009), and (c) Wang and Curry (2012)

In many cases, the trade‐off between SOC and BW is of great importance. Three concrete examples are provided for a better understanding of the importance of the trade‐off. Example 1 Masstige products

The SOC model is appropriate for a mass‐market product for which sales are more important. In contrast, the BW problem suits a niche‐market product that aims to provide high perceived prestige to customers. Masstige is an emerging kind of product located between the two types of products. The masstige products focus on a broader customer group (larger SOC) than its niche‐market competitors while having a higher perceived prestige (larger BW) than the mass‐market ones (Truong et al., 2009). For example, Coach is a company that provides masstige products. The company positions its leather goods between the traditional luxury goods, such as Gucci, and the mass‐market goods, such as Mossimo at Target (Silverstein & Fiske, 2003). A good trade‐off between SOC and BW can help companies position their masstige products better.

Example 2 Town library design, Gupta & Kohli, 1990

Consider a township planning to build a new library. Suppose that the residents can still use the current library, and there are three feasible, new alternatives (differing in the collection, number of books, and video services, etc.). Suppose there are five segments in the market, with relative sizes and utility as shown in Table 1.

Like many decisions in the public sector, decision makers care about both social benefit (measured by BW) and the percentage of people who benefited from the new alternative (measured by SOC). If BW is applied as a single criterion, the township should choose option A. However, less than one‐half of the population will be better off with A. Option B, on the other hand, maximizes SOC, but sacrifices a large portion of social benefits. In this case, option C might be a better choice because it makes a good trade‐off between the two objectives.

Example 3 Profit‐market share trade‐off in Intel

The market share profit trade‐off is key to product design choices in many industries. For example, Li et al. (2019) noted that “at Intel, for example, senior management constantly shifts discussion between profit maximization and market share expansion. On the one hand, the profit‐maximizing pricing solution may not meet the firm's ambition on market share; on the other hand, the market share‐maximizing prices reduce profit margins to nil, which is also far from ideal.” As BW indicates the social surplus left in the market, the trade‐off between market share and profit is also partly determined by the trade‐off between SOC and BW.

TABLE 1

Detail utility of the town library design example, Gupta and Kohli (1990)

		Alternatives
Segment	Number of residence	A	B	C	D a
1	7	10	7	1	6
2	28	5	3	4	2
3	32	6	8	7	9
4	6	11	3	12	4
5	14	4	6	7	5
Objectives	BW	1.77	0.56	1.52	0
	SOC	0.47	0.56	0.55	0

^a

The current library (status quo).

To answer the research question, we propose a distributionally robust optimization (DRO) model for SOC maximization using an uncertainty set based on Wasserstein distance, and another BW maximization model based on the notion of winsorization, that is, truncating the outliers (abnormally high values) at a certain threshold, so that the aggregate BW calculation will not be heavily influenced by some abnormal values. The former provides robustness against distributional deviation of customers' utility for SOC maximization, and the latter provides robustness against outliers for BW maximization. Our analysis of the two models yields the following important insights:

Equivalence. We prove that given any extent of robustness for the DRO model of the SOC problem, there exists a winsorized BW model sharing the same optimal solution. Furthermore, the optimizer of a more robust DRO model for the SOC problem (i.e., larger uncertainty set) is optimal to a less robust winsorized model (a higher threshold for truncation).

Trade‐off between SOC and BW. Previous robust product design models focus on a single objective (either SOC or BW). We find that adjusting the extent of robustness also achieves the trade‐off between the two objectives. The solution to a more robust DRO model of the SOC problem tends to have a higher BW compared to that of a less robust model. Conversely, the solution to a BW model with a greater extent of winsorization (higher threshold) has a higher SOC as well.

Extension of the first‐choice model. The classical SOC model applies the first‐choice assumption to characterize customer choice, that is, customers are classified into those who buy and those who do not buy, according to the sign of their consumer surplus. The robust counterpart allows us to introduce the third segment with probabilistic choice for the SOC problem. This corresponds to identifying outliers in the BW problem.

Solution approach with lower complexity. Based on the equivalence, we develop an approach that first solves the winsorized BW model for the different extents of winsorization and then picks the optimal solutions for the DRO model of the SOC problem. Because the winsorized BW model is easier to solve compared to the DRO model for the SOC problem, the proposed approach reduces the complexity of solving the proposed DRO model.

The remainder of this paper is organized as follows. Section 2 reviews previous literature. Section 3 provides a formal statement of the SOC and BW problems and a discussion on the necessity of the robustness and trade‐off between the two objectives. Section 4 proposes the DRO model of SOC maximization and the winsorized BW model. Section 5 presents the analytical results. Section 6 presents numerical experiments with synthetic and real‐life data. Finally, conclusions and future research directions are outlined in Section 7. For the briefness of reading, we furnish all the proofs in the Supporting Information (EC.3). Notations

Throughout the paper, scalar values are denoted by the standard letter x. Vectors are denoted by the bold letter x . The sets are denoted by the calligraphic capital letter

$\mathcal{X}$

. For any integer N, [N] is shorthand for the set

$\{1,\text{…},N\}$

. We denote random variables by letters with tilde notation and the corresponding sampled observation by those with hat notation. For example,

$\widehat{u}$

represents a sampled observation of random variable

$\stackrel{\sim }{u}$

,

${\{u\}}^{+}$

represents the maximum of u and 0, and

$\mathbf{1}\{·\}$

is an indicator function that equals 1 if the condition in the braces holds, and 0 otherwise.

LITERATURE REVIEW

The classical product design models proposed in the literature can be divided into first‐choice or probabilistic‐choice models. The first‐choice model assumes that the individual customers deterministically choose their most preferred product with the highest utility (Belloni et al., 2008; Camm et al., 2006; Wang et al., 2009). The choices of individual customers are aggregated to calculate objectives in an empirical risk optimization way; that is, sample statistics of objectives are applied to evaluating products. The main focal point of these papers is computational efficiency. For example, Bertsimas and Mišić (2019) provided a novel mixed‐integer optimization formulation for the first‐choice product design model and show that the formulation is stronger than the classic formulations in Belloni et al. (2008) and McBride and Zufryden (1988). In contrast, the probabilistic choice model considers a random choice, where the probability is a prespecified function of utility. Given the choice probability, the expectation of objectives can be calculated. Probabilistic choice models with different prespecified functions are applied in previous literature, such as the multinomial logit model (Chen & Hausman, 2000), the share‐of‐surplus model (Kraus & Yano, 2003), and the general attraction model (Schön, 2010a, 2010b). The first‐choice model is the most widely used in product (line) design literature. Our model extends the first‐choice model by a Wasserstein DRO approach.

Both first and probabilistic choice models assume that the utility and choice probabilities can be estimated precisely. However, estimation errors in the utilities and the probabilities are inevitable in practice (Bertsimas & Mišić, 2016; Wang & Gutierrez, 2022). Therefore, some researchers studied how to offer robustness against uncertainty in utility. Camm et al. (2006) repeatedly solved the classic models and selected the most frequently occurring solutions as the robust ones for the single‐product SOC problem. Belloni et al. (2008) implemented a similar procedure in a revenue‐maximizing product‐line design problem to test the robustness of solutions obtained by different solving approaches. Wang et al. (2009) conducted a similar robustness test for the SOC product‐line design problem.

Although emphasizing the importance of customer preference uncertainty, these studies do not explicitly model the uncertainty. Recent papers paid attention to robust models for the product design problem (Bertsimas & Mišić, 2016; Wang & Curry, 2012; Wang & Gutierrez, 2022). Bertsimas and Mišić (2016) proposed a robust revenue‐maximizing product‐line design model, which maximizes the worst‐case revenue over a structural uncertainty set containing choice models with different types and parameters. The above paper differs from our paper in two aspects. First, the model focuses on selecting the products from a prespecified candidate set while our model builds the products by combining multiple attributes. Second, their model focuses more on the probabilistic choice model while our model extends the first‐choice model. Compared to Bertsimas and Mišić (2016), Wang and Curry (2012) and Wang and Gutierrez (2022) are more closely related to our paper because both their models are based on the first‐choice model and maximize the SOC. As shown in Section 6.1.3, both their models can be nested into the same max–min framework as ours. The major difference between the two papers and ours is the uncertainty sets adopted. Specifically, Wang and Curry (2012) maximized the worst‐case SOC with an event‐wise uncertainty set proposed by Chen et al. (2020) while Wang and Gutierrez (2022) maximized the worst‐case SOC with scenario‐based uncertainty set.

Moreover, our paper distinguishes itself by showing the relationship between robustness against customer preference uncertainty for the SOC problem and another widely studied objective, that is, BW (Balakrishnan & Jacob, 1996; Green & Krieger, 1985; Kohli & Sukumar, 1990). The result is general. It enhances the insights from the numerical results obtained in Camm et al. (2006) and Wang et al. (2009) and holds for the model proposed by Wang and Curry (2012) and Wang and Gutierrez (2022) as we will show by the numerical experiments.

PROBLEM STATEMENT

We extend the conventional product design problem based on the first‐choice model, which is widely studied in previous literature (Balakrishnan & Jacob, 1996; Camm et al., 2006; Kohli & Sukumar, 1990; Shi et al., 2001; Wang & Curry, 2012). We provide the problem statement, benchmark models, and a discussion on the disadvantages of the benchmark models in this section.

First choice based framework

Although different in the objectives, the product design frameworks based on the first‐choice model have threefold primitives.

Product representation. A product is considered to be composed of several attributes. Let A denote the total number of attributes. The specification of the product is described by the levels selected for the attributes. We use

${\mathcal{L}}_a$

to denote the candidate set of the level for each attribute

$a\in [A]$

. For briefness, let

$\mathcal{L}={\cup }_{a\in [A]}{\mathcal{L}}_a$

denote the whole candidate set for all attributes. Therefore, the design of a product is to assign values for a vector of binary variables,

$\mathit{x}=\{x_1,\text{…},x_{|\mathcal{L}|}\}$

, where

$x_l=1$

if level l is selected, and 0 otherwise.

Preference measurement. A sample of representative customers is drawn from the entire population. Let N denote the total number of sampled customers. We use the partworth utility of level

$l\in \mathcal{L}$

for customer

$n\in [N]$

, denoted as

${\widehat{u}}_{nl}$

, to represent customer preference about the level, which is estimated according to her/his response to the CA survey. We refer the readers to the Supporting Information (EC.1.2) for the detail of data collection and utility estimation in CA. The key assumption in the CA is that the utility of a product x is the summation of the partworth utilities of selected levels; that is, the utility of product x for customer n can be computed by

${\sum }_{l\in \mathcal{L}}{\widehat{u}}_{nl}{x}_l$

. In addition, let

${\widehat{h}}_n$

denote the hurdle utility of customer n to keep the status quo or choose the outside option, which can be estimated by incorporating the outside options in the CA survey.

Choice modeling. We follow the first‐choice model to characterize customer choice from the offered product and the outside option. The offered product will be selected if it has higher utility than the outside option. Define consumer surplus of customer n as

${\widehat{c}}_n(\mathit{x})={\sum }_{l\in \mathcal{L}}{\widehat{u}}_{nl}{x}_l-{\widehat{h}}_n$

. Accordingly, customer n will choose the product if

${\widehat{c}}_n(\mathit{x})>0$

; otherwise, she/he will choose the outside option. In this paper, we refer to customers with nonnegative surpluses as buyers.

Benchmark models

Based on the primitives, previous scholars have proposed several prescriptive models to find a product x in a feasible set, denoted as

$\mathcal{X}$

, such that some managerial objectives are maximized. In the main text, we focus on a simple form,

$\mathcal{X}=\{\mathit{x}|{\sum }_{l\in {\mathcal{L}}_a}{x}_l=1,x_l\in \{0,1\},\forall a\in [A]\}$

, which only constrains that only one level should be selected for each attribute. Note that our model and results can be easily extended to study product‐line design problems with constraints often encountered in practice. We refer the readers to the Supporting Information (EC.1.2) for detail.

We focus on two widely studied objectives, that is, SOC and BW (Balakrishnan & Jacob, 1996; Camm et al., 2006; Green & Krieger, 1985; Kohli & Sukumar, 1990; Shi et al., 2001). The former is the percentage of buyers, while the latter represents the total surplus collected by all buyers.

Generally, the decision makers aim to maximize the managerial objectives over the entire customer population. However, in practice, it is impossible to obtain information about the entire population. A straightforward way to overcome this difficulty is to maximize the objective over a sample of representative customers instead. Let

$$\begin{array}{c}\hfill {\widehat{\mathbb{P}}}_N=\frac{1}{N}\sum _{n\in [N]}\delta ({\widehat{\mathit{u}}}_n,{\widehat{h}}_n)\end{array}$$

1

denote the empirical distribution of customers' partworth and hurdle utilities, where

$\delta ({\widehat{\mathit{u}}}_n,{\widehat{h}}_n)$

denotes the Dirac distribution concentrating unit mass at

$({\widehat{\mathit{u}}}_n,{\widehat{h}}_n)$

. The classic SOC model (Camm et al., 2006; Shi et al., 2001) is formulated as follows:

2

and the BW model (Balakrishnan et al., 2004; Dobson & Kalish, 1993; Green & Krieger, 1985) is given as

3

Note that the formulations provided in the main text are to show the relationship among the models. The mixed‐integer linear programming (MILP) reformulations of the models are presented in the Supporting Information (EC.1.4) for computation. Readers may find that the formulations above are slightly different from those proposed in previous literature. We discuss the equivalence between the formulations and previous formulations in the Supporting Information (EC.1.3).

Issues with the benchmark models

This section provides a simple example to illustrate the drawbacks of models SOC‐D and BW‐D. Example 4

Consider a problem where decision makers need to select one product from a product set

$\{a,b,c\}$

facing a sample of five customers, that is, 1,2,3,4,5. The data are presented in Table 2.

The rows under the case of “Observed” report the observed surplus of the three products for the five customers. According to observed consumer surplus, model SOC‐D selects product a while model BW‐D chooses product c. These two products are “extreme” because product a has a very small average BW (0.1), whereas product c has the lowest SOC (0.2). The example illustrates the drawback of solely considering SOC and BW, which has already been discussed in Gupta and Kohli (1990). Product b is a more desirable product for decision makers who care about both objectives and hope to strike a balance between the two objectives.

The fourth to ninth rows are used to show the vulnerability of SOC and BW maximizers to different types of customer preference uncertainty. The SOC maximizer, product a in this case, is vulnerable to distributional deviation in the consumer surplus. In the “Deviate” case, where the ground truth surplus deviates downward uniformly by 0.2 from the observed cases, the SOC of product a is reduced to 0. The BW maximizer, product c, is sensitive to what is known as outliers in statistics (Grubbs, 1969; Maddala, 1992). As shown in the “Outlier” case, both the BW and SOC of product c are reduced to 0 when customer 1 drops out. Because product b shows greater robustness in both “Deviate” and “Outlier” cases, it may be preferred by the decision makers.

TABLE 2

Specific data for Example 1

		Customer					Objective
Case	Product	1	2	3	4	5	SOC	BW
	a	0.1	0.1	0.1	0.1	0.1	1.00	0.10
Observed	b	−1.0	2.0	2.0	2.0	2.0	0.80	1.60
	c	10.0	−1.0	−1.0	−1.0	−1.0	0.20	2.00
	a	−0.1	−0.1	−0.1	−0.1	−0.1	0.00	0.00
Deviate	b	−1.2	1.8	1.8	1.8	1.80	0.8	1.44
	c	9.8	−1.2	−1.2	−1.2	−1.2	0.20	1.96
	a	–	0.1	0.1	0.1	0.1	1.00	0.08
Outlier	b	–	2.0	2.0	2.0	2.0	1.00	1.60
	c	–	−1.0	−1.0	−1.0	−1.0	0.00	0.00

To address the impact of customer preference uncertainty, we propose a DRO model for the SOC problem and a winsorized model for the BW problem in Section 4. Analytical results in Section 5 show that product b is selected by both models when the parameters are set properly. Therefore, the results indicate that both models are capable of providing robustness against both “Deviate” and “Outlier” cases, which is closely related to the trade‐off between SOC and BW.

MODEL FORMULATION

The DRO counterpart for model SOC‐D

As shown in the “Deviate” case in Example 2, model SOC‐D is vulnerable to a distributional deviation between the empirical and true distributions. The Wasserstein DRO framework is one of the most popular frameworks to offer robustness against such deviation and has many desirable properties (Esfahani & Kuhn, 2018; Kuhn et al., 2019). We adopt this framework to the SOC problem; that is, we maximize the worst‐case SOC in a Wasserstein uncertainty set. The set is centered at the empirical distribution and contains all the distributions that deviate from the empirical distribution by Wasserstein distance of no more than a given threshold. The Wasserstein distance is defined as follows. Definition 1 Wasserstein distance; Esfahani & Kuhn, 2018

4

where Ξ² is the support of ξ and ξ ₀ and

${\parallel ·\parallel }_p$

represents the

$l_p$

norm.

Accordingly, the uncertainty set is then defined as

5

where a nonnegative ε denotes the maximal Wasserstein distance between the distributions in set

$\mathcal{D}$

and the empirical distribution. The DRO model of SOC maximization is as follows:

6

Note that although we maximize the worst‐case SOC here, our target is to provide a statistical guarantee to the SOC under the true distribution. Many results on how to adjust ε to achieve a desirable guarantee can be found in previous literature (Esfahani & Kuhn, 2018; Zhao & Guan, 2018). Furthermore, the framework is flexible enough to reflect decision makers' different attitudes toward uncertainty. The conservatism of model SOC‐DRO increases in ε. Therefore, a designer who believes that the partworth utilities fully reflect the preference of the customer population can apply the model with

$\epsilon =0$

, which maximizes the average SOC over the sample, whereas a conservative designer who worries about estimation error in the partworth utility can apply the model with a large ε.

Winsorized BW model

The “Outlier” case in Example 2 shows that the BW maximization is sensitive to a customer with an extremely high consumer surplus, which is known as outliers in statistics. Winsorization is a common statistic technique to deal with outliers, which replaces the extreme values with a prespecific value. The technique is applied in many fields, such as finance (Berg‐Jacobsen & Tran, 2021; Khan & Fahim, 2021), biometrics (Lan et al., 2022; Li et al., 2021), and psychology (Anderson et al., 2022; Sales et al., 2021), and is commonly incorporated the mainstream statistic software like SPSS, ² STATA, ³ and scipy ⁴ package for Python. In this paper, we adopt this technique to formulate a winsorized BW maximization model as follows:

$$\begin{array}{c}\hfill \text{[BW-R]}\underset{\mathit{x}\in \mathcal{X}}{\max }{\mathbb{E}}_{{\widehat{\mathbb{P}}}_N}(\min ({\{\stackrel{\sim }{c}(\mathit{x})\}}^{+},c)).\end{array}$$

7

Intuitively speaking, the practical idea behind model BW‐R is to forbid the phenomenon—“the squeaky wheel gets the grease.” An upper limit of c is a threshold to distinguish between ordinary and extremely high consumer surplus. With this term, no matter how high the consumer surplus a customer has, his surplus is winsorized to c.

This framework is also flexible to decision makers with different attitudes toward uncertainty. As c decreases, the winsorized BW model is less sensitive to outliers because customers with a higher consumer surplus than c contribute less to the objective function. With a large enough c, the BW model is equivalent to the classic BW model.

ANALYTIC RESULTS

In this section, we provide the condition where models SOC‐DRO and BW‐R are equivalent. Motivated by the equivalence, we show how the models achieve a trade‐off between SOC and BW and develop an approach to get the optimal solution of the proposed SCO‐DRO model with lower complexity.

The relationship between models SOC‐RDRO and BW‐R

As a foundation of this section, we first present a tractable reformulation of model SOC‐DRO:

8

9

10

11

where λ is a standard Lagrangian dual variable of the constraint in set (5);

$R=\epsilon {(A+1)}^{\frac{p-1}{p}}$

;

$y_n$

is an indicator variable equal to 1 if customer n is a buyer, and 0 otherwise;

$s_n$

is an auxiliary variable equal to

$\min \{y_n,-\lambda {\widehat{c}}_n(\mathit{x})y_n\}$

.

As shown in Definition 1 and Equation (5), there are two parameters, distance ε and norm p, to control the extent of the robustness of model SOC‐RDRO. Model SOC‐RDRO indicates that the influence of the two parameters can be synthesized by a single metric R, which makes it easy to compare the extent of the robustness of models SOC‐RDRO using different parameters. Therefore, we investigate the effect of R instead of that of ε and p independently in the rest of this paper. Note that R increases in both ε and p, and therefore a large R indicates a more conservative model SOC‐RDRO.

The core of the difference between models SOC‐RDRO and SOC‐D is the introduction of dual variable λ. When

$\lambda =-\infty$

, constraint (10) is ineffective and therefore

$s_n=y_n$

. Accordingly, the contribution of customer n is a step function of

${\widehat{c}}_n(\mathit{x})$

,

$\forall n\in [N]$

, as follows:

12

A graphic demonstration of Equation (12) is given in Figure 2a. The function is the same as the one applied to evaluate the contribution of customer n in model SOC‐D. When

$R=0$

, setting

$\lambda =-\infty$

is optimal because in this situation,

$\lambda =-\infty$

makes constraint (10) ineffective when

${\widehat{c}}_n(\mathit{x})\ge 0$

, which broadens the feasible set, and does not influence the objective value through the term

$\lambda R$

. This indicates model SOC‐RDRO with

$R=0$

is equivalent to model SOC‐D.

FIGURE 2

Comparison between

$y_n$

in model SOC‐D and

$s_n$

model SOC‐RDRO: (a) SOC‐D and (b) SOC‐RDRO

For

$\lambda \in (-\infty ,0)$

, the contribution of customer n is a piecewise linear function

${\widehat{c}}_n(\mathit{x})$

,

$\forall n\in [N]$

, as follows:

13

A graphic demonstration of Equation (13) in Figure 2b. The function is of the same shape as the one in model BW‐R. Specifically, when

${\widehat{c}}_n(\mathit{x})$

is less than a threshold, customer n's contribution to the objective is proportionate to

${\widehat{c}}_n(\mathit{x})$

; and when

${\widehat{c}}_n(\mathit{x})$

exceeds the threshold, customer n's contribution to the objective is equal to the threshold. This motivates us to explore the relationship between models SOC‐RDRO and BW‐R.

Let

${\lambda }_R^{\ast }$

denote the optimal λ of model SOC‐RDRO with parameter R and

$\underline{c}=\min \{{\widehat{c}}_n(\mathit{x})|{\widehat{c}}_n(\mathit{x})>0,\mathit{x}\in \mathcal{X},n\in [N]\}$

denote the minimal positive consumer surplus overall

$\mathit{x}\in \mathcal{X}$

and

$n\in [N]$

. Proposition 1 shows that for model SOC‐RDRO with any

$R\in [0,\infty )$

, there exists a model BW‐R sharing the same optimal x . Proposition 1 Equivalence condition between BW‐R and SOC‐RDRO

Models SOC‐RDRO and BW‐R share the same optimizer, when

$R=0$

and

$c\in (0,\underline{c}]$

;

$R\in (0,{\max }_{\mathit{x}\in \mathcal{X}}\frac{{\sum }_{n\in [N]}{\{{\widehat{c}}_n(\mathit{x})\}}^{+}}{N})$

and

$c=-\frac{1}{{\lambda }_R^{\ast }}\ge \underline{c}$

;

$R\in [{\max }_{\mathit{x}\in \mathcal{X}}\frac{{\sum }_{n\in [N]}{\{{\widehat{c}}_n(\mathit{x})\}}^{+}}{N},\infty )$

and

$c=0$

.

Note that

${\max }_{\mathit{x}\in \mathcal{X}}\frac{{\sum }_{n\in [N]}{\{{\widehat{c}}_n(\mathit{x})\}}^{+}}{N}$

, that is, the maximal BW over all

$\mathit{x}\in \mathcal{X}$

, is the upper limits of R. Model SOC‐RDRO with

$R\ge {\max }_{\mathit{x}\in \mathcal{X}}\frac{{\sum }_{n\in [N]}{\{{\widehat{c}}_n(\mathit{x})\}}^{+}}{N}$

is meaningless because

${\lambda }_R^{\ast }$

equals to 0 in this situation and therefore,

$\forall \mathit{x}\in \mathcal{X}$

, the corresponding objective values are 0. Furthermore, model BW‐R with

$c=0$

is meaningless for the same reason. Therefore, we limit our analysis within the range of

$R\in [0,{\max }_{\mathit{x}\in \mathcal{X}}\frac{{\sum }_{n\in [N]}{\{{\widehat{c}}_n(\mathit{x})\}}^{+}}{N})$

for model SOC‐RDRO and

$c\in (0,\infty )$

for model BW‐R.

Another keynote is that for R close enough to

${\max }_{\mathit{x}\in \mathcal{X}}\frac{{\sum }_{n\in [N]}{\{{\widehat{c}}_n(\mathit{x})\}}^{+}}{N}$

, model SOC‐RDRO finds the BW maximizer, that is, the optimal solution of model BW‐D (model BW‐R with large enough c). Let

${\mathit{x}}_B^{\ast }$

denote the optimal solution of model BW‐D. Corollary 1 presents the condition where the optimal solution of model SOC‐RDRO is

${\mathit{x}}_B^{\ast }$

.

Corollary 1

The optimal solution of model SOC‐RDRO with

$R\in [{\max }_{\mathit{x}\in \mathcal{X}\setminus \{{\mathit{x}}_B^{\ast }\}}\frac{{\sum }_{n\in [N]}{\{{\widehat{c}}_n(\mathit{x})\}}^{+}}{N},{\max }_{\mathit{x}\in \mathcal{X}}\frac{{\sum }_{n\in [N]}{\{{\widehat{c}}_n(\mathit{x})\}}^{+}}{N})$

is

${\mathit{x}}_B^{\ast }$

.

Furthermore, we illustrate the monotonicity of

${\lambda }_R^{\ast }$

in R in Proposition 2. Proposition 2

${\lambda }_R^{\ast }$

is nondecreasing in R.

According to Proposition 2,

$-\frac{1}{{\lambda }_R^{\ast }}$

is also nondecreasing in R. As illustrated in Proposition 1, model SOC‐RDRO with

$R=0$

and

$R\in [0,{\max }_{\mathit{x}\in \mathcal{X}}\frac{{\sum }_{n\in [N]}{\{{\widehat{c}}_n(\mathit{x})\}}^{+}}{N})$

shares the same optimizer with model BW‐R with

$c\in (0,\underline{c}]$

and

$c=-\frac{1}{{\lambda }_R^{\ast }}\ge \underline{c}$

, respectively. Therefore, model SOC‐RDRO with a larger R shares the same optimizer with model BW‐R with a larger c. Because a larger c corresponds to a smaller extent of winsorization, it indicates that the notions of the robustness of model SOC‐RDRO are dual to that of the winsorization of model BW‐R. Thus, the optimal product of model SOC‐DRO has a double meaning. Specifically, the optimal solution of model SOC‐DRO with a larger R is a more robust solution against distributional deviation for the SOC problem as well as a solution that is more sensitive to the outliers with respect to the BW maximization. It is worth noting that model SOC‐DRO with R equal or close enough to 0 and

${\max }_{\mathit{x}\in \mathcal{X}}\frac{{\sum }_{n\in [N]}{\{{\widehat{c}}_n(\mathit{x})\}}^{+}}{N}$

(corresponding to model BW‐R with c close to 0 and large enough, respectively) obtains the SOC and BW maximizers, respectively. This indicates that the BW maximizer is the most robust solution for the SOC problem while the SOC maximizer is the solution least sensitive to the outliers for the BW problem. Remark 1 A new classification scheme for the SOC problem

The difference between (12) and (13) indicates not only the relationship between models SOC‐RDRO and BW‐R but also a different classification scheme compared to that under model SOC‐D. As shown in Figure 2, the new scheme further classifies the original buyers into two types. An intuitive probabilistic explanation of the scheme is that the buyers with surpluses lower than a threshold will purchase the offered product with a probability proportional to the scale of their surplus. This indicates that the underlying choice model of model SOC‐RDRO is more flexible than the first‐choice model under the classic SOC model, which cannot characterize probabilistic choice.

With the probabilistic choice modeling, the new scheme can overcome two technical difficulties in the classic SOC model. The first difficulty is the classification of customer with

${\widehat{c}}_n(\mathit{x})=0$

. As an extreme example, when there exists a

$\mathit{x}\in \mathcal{X}$

delivering the same utility as the outside option, that is,

${\widehat{c}}_n(\mathit{x})=0,\forall n\in [N]$

, the product is optimal if customers with 0 surpluses are classified as buyers and has 0 objective value otherwise. In contrast, the product has 0 objective values in our model in both situations. The second is the abrupt change in customer choice. It is hard to convince the practitioners to believe that customers with a little difference in consumer surplus, for example, one customer with 0.1 surplus and the other with −0.1, would have such a sudden turn in their choice as modeled in the first‐choice model. In contrast, the choice probability of a customer in our model is a continuous function of the surplus and does not have abrupt change.

Trade‐off between SOC and BW

As we illustrated in the last section, model SOC‐RDRO with R equal or close enough to 0 and

${\max }_{\mathit{x}\in \mathcal{X}}\frac{{\sum }_{n\in [N]}{\{{\widehat{c}}_n(\mathit{x})\}}^{+}}{N}$

obtains the SOC and BW maximizers, respectively. In this section, we show that the transition from SOC maximization to BW maximization is a gradual process. Specifically, as R increases, model SOC‐RDRO focuses less on SOC and more on BW, which achieves the trade‐off between SOC and BW. We first present a numerical instance to illustrate the trade‐off and then provide analytical results to show how the trade‐off is achieved.

Figure 3 illustrates the trade‐off observed in a numerical instance. The figure shows that model SOC‐RDRO selects the same products as models SOC‐D and BW‐D when the degree of conservatism is least and highest; as R increases, the BW of the selected product becomes higher while the corresponding SOC decreases.

FIGURE 3

Relationship between models SOC‐D, BW‐D, and the proposed DRO model: (a) SOC comparison and (b) BW comparison

The trade‐off is due to the selection of λ. For the case of

$R>0$

, Proposition 3 presents the closed‐form solution of λ that maximizes the objective function (8) for a given

$\mathit{x}\in \mathcal{X}$

. Let

${\lambda }^{\ast }(\mathit{x},R)$

denote the λ such that the objective value of a given x is maximized. Proposition 3

For model SOC‐RDRO with

$R>0$

,

${\lambda }^{\ast }(\mathit{x},R)$

can be expressed as

14

where

$$n^{\ast }=\min \left\{n^{\prime }|R\le \frac{{\sum }_{n=1}^{n^{\prime }}{\{{\widehat{c}}_{(n)}(\mathit{x})\}}^{+}}{N}\right\},$$

15

and

${\widehat{c}}_{(N)}(\mathit{x})\ge \cdots {\widehat{c}}_{(2)}(\mathit{x})\ge {\widehat{c}}_{(1)}(\mathit{x})$

are order statistics of

$\{{\widehat{c}}_n(\mathit{x}),\forall n\in [N]\}$

.

Proposition 3 shows that given x , the corresponding optimal λ can be found through a greedy heuristic. Specifically,

$-\frac{1}{\lambda }$

is the

$n^{\ast }$

th smallest consumer surplus where

$n^{\ast }$

is the smallest integer such that

$\frac{{\sum }_{n=1}^{n^{\ast }}{\{{\widehat{c}}_{(n)}(\mathit{x})\}}^{+}}{N}\ge R$

. To better illustrate Proposition 3, we provide a simple example as follows. Example 5

Consider a product x with

$N=5$

and the surplus value of the customers in nondecreasing order is shown in Table 3. If

$R=0.5$

,

$(n^{\ast })=(4)$

because

$\frac{{\sum }_{(n)=1}^{(3)}{\{{\widehat{c}}_{(n)}(\mathit{x})\}}^{+}}{N}=0.4\le 0.5\le \frac{{\sum }_{(n)=1}^{(4)}{\{{\widehat{c}}_{(n)}(\mathit{x})\}}^{+}}{N}=1$

. Consequently,

${\widehat{c}}_{(n^{\ast })}(\mathit{x})={\widehat{c}}_{(4)}(\mathit{x})=3$

and

${\lambda }^{\ast }(\mathit{x},R)=-\frac{1}{3}$

. When

$R=2$

where

$\frac{{\sum }_{(n)=1}^{(5)}{\{{\widehat{c}}_n(\mathit{x})\}}^{+}}{N}=1.8\le 2$

,

${\lambda }^{\ast }(\mathit{x},R)=0$

.

TABLE 3

Data for Example 5

Ordered index	(1)	(2)	(3)	(4)	(5)
${\widehat{c}}_n(\mathit{x})$	−1	−2	2	3	4
Cumulative of ${\displaystyle \frac{{\{{\widehat{c}}_n(\mathit{x})\}}^{+}}{N}}$	0	0	0.4	1	1.8

Proposition 3 shows a twofold influence of R on selecting the optimal solution.

First, model SOC‐RDRO requires the BW of the selected product to exceed R. For given x ,

${\lambda }^{\ast }(\mathit{x},R)$

is equal to 0 if the BW of x , that is,

$\frac{{\sum }_{n\in [N]}{\{{\widehat{c}}_n(\mathit{x})\}}^{+}}{N}$

is less than R and consequently, the corresponding objective function is equal to 0. In contrast, the product with BW higher than R has an objective value greater than 0. Therefore, the optimal solution of model SOC‐RDRO must be selected from those with BW higher than R. We summarize the above discussion in Corollary 2. Let x _R denote the optimal solution of model SOC‐RDRO with a given R. Corollary 2

When

$R\in [0,{\max }_{\mathit{x}\in \mathcal{X}}\frac{{\sum }_{n\in [N]}{\{{\widehat{c}}_n(\mathit{x})\}}^{+}}{N})$

, the BW of x _R must be higher than R.

Second, model SOC‐RDRO maximizes the percentage of a smaller subset of buyers as R increases. The following proposition shows that instead of maximizing the percentage of buyers, model SOC‐RDRO maximizes the percentage of buyers with consumer surplus higher than

$-\frac{1}{{\lambda }^{\ast }(\mathit{x},R)}$

. Proposition 4 Maximization of the percentage of buyers with high surplus

$$J(\mathit{x},R)\le J({\mathit{x}}_R,R),\forall \mathit{x}\in \mathcal{X},$$

16

where

$J(\mathit{x},R)=\frac{|\{n|{\widehat{c}}_n(\mathit{x})\ge -\frac{1}{{\lambda }^{\ast }(\mathit{x},R)}\}|}{N}$

is the percentage of buyers with surplus higher than

$-\frac{1}{{\lambda }^{\ast }(\mathit{x},R)}$

.

Recall that as shown in Proposition 3,

$-\frac{1}{{\lambda }^{\ast }(\mathit{x},R)}={\widehat{c}}_{(n^{\ast })}(\mathit{x})$

where

${\widehat{c}}_{(n^{\ast })}(\mathit{x})$

is the

$n^{\ast }$

th smallest consumer surplus and

$n^{\ast }$

is the smallest integer such that

$\frac{{\sum }_{n=1}^{n^{\ast }}{\{{\widehat{c}}_{(n)}(\mathit{x})\}}^{+}}{N}\ge R$

. Therefore,

$\forall \mathit{x}\in \mathcal{X}$

,

$-\frac{1}{{\lambda }^{\ast }(\mathit{x},R)}={\widehat{c}}_{(n^{\ast })}(\mathit{x})$

increases as R increases, and consequently,

$\{n|{\widehat{c}}_n(\mathit{x})\ge -\frac{1}{{\lambda }^{\ast }(\mathit{x},R)}\}$

is a smaller subset of all buyers.

To synthesize, as R increases, model SOC‐RDRO puts a higher lower bound on the BW of the selected product while focusing on a smaller subset of buyers, which leads to the trade‐off between SOC and BW as shown in Figure 3. Thus, the proposed model can help practitioners make a better trade‐off between the two objectives. For example, for the design of a masstige product as illustrated in Example 1, our model can provide several recommendations with different trade‐off positions between the two objectives. As the robust parameter R increases from 0 to ∞, the recommendation gradually turns from a mass product to a product that focuses on a smaller range of customers with high surplus value. Then, the practitioners can find their sweet spot between sales and perceived prestige from the recommendations according to their preferences. Furthermore, the trade‐off also guides a decision maker with a single objective. For a decision maker who aims to maximize SOC and wants to hedge against a greater degree of customer preference uncertainty, she/he should select a product with higher BW; similarly, for a decision maker who wants to release the sensitivity against outliers and maximize BW, she/he should search for a product with higher SOC.

An approach to solve model SOC‐RDRO

Although the product design is a long‐term decision, computational efficiency is of great importance when involving robustness. In practice, it is necessary to find the “proper” robust parameter R or a set of candidates with different extents of robustness to practitioners for further selection. This requires repeating solving model SOC‐RDRO with different R for an extensive number of times. Furthermore, it is prudent to apply cross‐validation to further validate the performance of the models with different R (Esfahani & Kuhn, 2018), that is, for a given R, model SOC‐RDRO needs to be solved multiple times with different samples. Therefore, model SOC‐RDRO may be solved thousands of times to get a satisfying recommendation for the practitioners.

As model BW‐R is easier to solve compared to model SOC‐RDRO, the equivalence between the two models indicates an approach to enhance the computational efficiency, that is, getting the optimal solution of the model SOC‐RDRO through solving model BW‐R. The computational advantages of model BW‐R are twofold.

Lower complexity of evaluating the objective for a given product. Based on Proposition 3, determining the value

${\lambda }^{\ast }(\mathit{x},R)$

for a given x in model SOC‐RDRO requires sorting N customers according to their surplus. Therefore, for a given

$\mathit{x}\in \mathcal{X}$

, the complexity of evaluating the objective of model SOC‐RDRO is

$\mathcal{O}(N\log N)$

. In contrast, the computational complexity of solving model BW‐R is

$\mathcal{O}(N)$

because c is a parameter, and sorting is no longer required. Therefore, it is easier to solve model BW‐R when applying enumeration techniques, such as branch‐and‐bound or dynamic programming.

Smaller scale of the MILP reformulation. When applying off‐the‐shelf solvers, we need to further linearize the models to MILP because the constraints involve the multiplication of decision variables and are therefore nonconvex. This enlarges the number of variables and constraints. Table 4 compares the number of variables and constraints of the MILP formulation of models SOC‐RDRO and BW‐R as shown in the Supporting Information (EC.1.4). As shown in the table, due to the term

$N|\mathcal{L}$

, model SOC‐RDRO has a much larger scale compared to model BW‐R.

TABLE 4

Comparison of the scale of the models

Model	Variables	Constraints	Big‐M method required
SOC‐RDRO	$2N+|\mathcal{L}|+N|\mathcal{L}|+1$	$(3+2|\mathcal{L}|)N$	Yes
BW‐R	$2N+|\mathcal{L}|$	2N	No

However, we cannot directly replace model SOC‐RDRO with model BW‐R. On the one hand, the equivalence between models SOC‐RDRO and BW‐R holds only with the condition

$c=-\frac{1}{{\lambda }_R^{\ast }}$

and it is not possible to get

${\lambda }_R^{\ast }$

before solving model SOC‐RDRO. On the other hand, for

$c\in (0,\infty )$

, there may not exist an R such that the corresponding models SOC‐RDRO and BW‐R share the same optimizer. This is because

${\lambda }_R^{\ast }$

takes value only from a discrete set.

In the following, we show how to pick the optimal solutions for models SOC‐RDRO with different R from those of models BW‐R with different c. First, Proposition 5 shows the feature that distinguishes the optimal solutions of model SOC‐RDRO from other optimal solutions of model BW‐R. Let

$Q(c)$

and x _c denote the optimal objective value and the optimal solution of model BW‐R with a given c, respectively. Proposition 5 Applying model BW‐R to solve model SOC‐RDRO

For any ordered sequence,

$0<c_1\le \cdots \le c_J<\infty$

, satisfying

17

• (a)

  there exists

  $R\in [0,\infty )$
  such that
  ${\mathit{x}}_{c_j}$
  is optimal with respect to model SOC‐RDRO with R;

• (b)

  for any

  $R\in [0,\infty )$
  , if
  $\exists j\in [J-1]$
  such that
  ${\lambda }_R^{\ast }\in [-\frac{1}{c_j},-\frac{1}{c_{j+1}}]$
  , then
  $\frac{c_{j}Q(c_{j+1})-c_{j+1}Q(c_j)}{c_j-c_{j+1}}(c_j+\frac{1}{{\lambda }_R^{\ast }})-\frac{1}{{\lambda }_R^{\ast }}Q(c_j)=c_{j}Q(-\frac{1}{{\lambda }_R^{\ast }})$
  .

Condition (17) indicates that the curve connecting

$(-\frac{1}{c_j},\frac{Q(c_j)}{c_j})$

is a concave piecewise linear envelope of the curve

$\frac{Q(c)}{c}$

as a function of

$-\frac{1}{c}$

. It implies that the corresponding solutions to the intersection between the two curves are optimal for some R. Figure 4 provides a graphic illustration of Proposition 5. It indicates that we can obtain the optimal solutions of model SOC‐RDRO with different R through the following steps:

• (a)

  solving model BW‐R with different

  $c\in (0,\infty )$
  ;

• (b)

  finding upper concave piecewise linear envelop of the curve that plots

  $\frac{Q(c)}{c}$
  against
  $-\frac{1}{c}$
  and the intersections between the curves;

• (c)

  collecting the corresponding solutions of the intersections.

FIGURE 4

Applying concave piecewise linear envelop of

$\frac{Q(c)}{c}$

against

$-\frac{1}{c}$

to find

${\{c_j\}}_J$

After finding

${\{c_j\}}_J$

,

$\lambda =-\frac{1}{c_j}$

and

$\mathit{x}={\mathit{x}}_{c_j}$

are the optimal solution for some R. If practitioners do not care about the exact value of R and only want to find optimal x for the different extent of robustness, Propositions 1 and 2 indicate that the solution of model BW‐R with a larger

$c_j$

is optimal to model SOC‐RDRO with a larger R. In contrast, to build the relationship between R and the solutions, one can solve the following problem for a given R:

$$\underset{j\in [J]}{\max }-\frac{1}{c_j}R+\frac{Q(c_j)}{c_j}.$$

18

Then, if

$c_j,\forall j\in [J]$

, maximizes the above problem, then

$\lambda =-\frac{1}{c_j}$

and

$\mathit{x}={\mathit{x}}_{c_j}$

are the optimal solution for the given R. Remark 2 Generalization of the approach

The DRO models of many problems have the reformulation in the following format:

$$\begin{array}{c}\hfill \underset{\lambda <0,\mathit{x}\in \mathcal{X}}{\max }\lambda \epsilon +\frac{{\sum }_{n\in [N]}{s}_n(\mathit{x},{\widehat{\xi }}_n,\lambda )}{N},\end{array}$$

19

where

$s_n(\mathit{x},{\widehat{\xi }}_n,\lambda )$

is a function of

$(\mathit{x},{\widehat{\xi }}_n,\lambda )$

that decreases in λ;

${\widehat{\xi }}_n$

is the nth observation of random coefficients; and λ is the dual variable of the Wasserstein uncertainty set as denoted in this paper.

Note that solving model BW‐R is equivalent to solving model SOC‐RDRO with λ given in advance. The proof of Proposition 5 does not require properties specific to models SOC‐RDRO and BW‐R. As such, following the same idea, we can apply the following problem to get the optimal solution of problem (19):

$$\begin{array}{c}\hfill S(\lambda )=\underset{\mathit{x}\in \mathcal{X}}{\max }\frac{{\sum }_{n\in [N]}{s}_n(\mathit{x},{\widehat{\xi }}_n,\lambda )}{N};\end{array}$$

20

that is, the intersections of

$S(\lambda )$

and its upper concave envelope correspond to the optimal solutions of the original DRO model.

NUMERICAL EXPERIMENTS

We implemented extensive numerical experiments based on simulated and real‐life data to validate the effectiveness of the proposed models. Our numerical experiments mimic the product design procedure in practice, namely,

• (a)

draw sample customers from a population and collect their utilities;

(b)

establish models to select the “optimal” products;

(c)

launch the product to market and observe its market performance.

In steps (b) and (c), BW and SOC are calculated in‐sample (with respect to the sample) and out of sample (with respect to oracle distribution or testing samples) to evaluate the market performance, respectively. We denote the in‐sample and out‐of‐sample BW and SOC as

${\text{BW}}_{\text{is}}$

,

${\text{BW}}_{\text{os}}$

,

${\text{SOC}}_{\text{is}}$

, and

${\text{SOC}}_{\text{os}}$

, respectively. The experiments were conducted on a computer with Windows 10, 2.60 GHz Intel Core i7‐6700HQ CPU, and 8 GB RAM and coded with Python 3.6.4. Cplex 12.8.0 is applied when an off‐the‐shelf solver is required.

Simulated experiments

We generate simulated data based on a CA case for a new camera design composed of 8 attributes and 18 levels, whose partworth follows a multivariate normal distribution. The detail of the case is reported in Gilbride and Allenby (2004). We provide the mean vector and covariance matrix of the multivariate normal distribution in the Supporting Information (EC.2.1) for the convenience of reference. Following Camm et al. (2006), we simulate hurdle utility by fixing a benchmark product. We draw samples from the oracle distribution to establish the models. One hundred instances were generated for each sample size

$N\in \{10,30,300,3000\}$

.

Performance comparison across model SOC‐RDRO with different extents of robustness

This section presents the in‐sample and out‐of‐sample metrics to show the benefits of offering robustness to the SOC problem. Figure 5 plots the mean of the metrics (lines) and the tube between the 20th and 80th quantile of out‐of‐sample metrics (shaded regions) against R. In Figure 5, the leftmost node of the horizontal axis corresponds to model SOC‐D (model SOC‐RDRO with

$R=0$

).

FIGURE 5

In‐sample and out‐of‐sample SOC (left axis, solid line, and blue shaded region) and BW (right axis, dashed line, and green‐shaded region) with different sample sizes: (a)

$N=10$

, (b)

$N=30$

, (c)

$N=300$

, and (d)

$N=3000$

The in‐sample metrics reflect that the trade‐off between SOC and BW is achieved through adjusting R. Note that the deterioration at the right‐hand side of each figure is because R is over its upper limit. Before the deterioration, as R increases,

${\text{SOC}}_{\text{is}}$

decreases and

${\text{BW}}_{\text{is}}$

increases. This reflects the trade‐off between SOC and BW found in this paper. In contrast, both

${\text{SOC}}_{\text{os}}$

and

${\text{BW}}_{\text{os}}$

are improved. These improvements are summarized in two ways.

• Better average market performance. As shown in Figure 5, the average out‐of‐sample metrics of model SOC‐RDRO improve when the robustness was within an appropriate range. This indicates that the products selected by model SOC‐RDRO achieve better market performance on average.

• Insensitivity to customer preference uncertainty. As shown in Figure 5, the 20th to 80th quantile tubes for both

  ${\text{SOC}}_{\text{os}}$
  and
  ${\text{BW}}_{\text{os}}$
  shrink as R increases. The observation indicates that model SOC‐RDRO is more immune to customer preference uncertainty than model SOC‐D.

Furthermore, we observe better predictability of

${\text{SOC}}_{\text{is}}$

to

${\text{SOC}}_{\text{os}}$

, especially when the sample size is small. As shown in Figure 5, the gaps between the

${\text{SOC}}_{\text{is}}$

and

${\text{SOC}}_{\text{os}}$

of model SOC‐D reduce the extent of overoptimism in the prediction of

${\text{SOC}}_{\text{os}}$

. In practice, overoptimistic prediction of

${\text{SOC}}_{\text{os}}$

leads to devastating results, such as investment in an unpopular product or overpurchase of raw material. These devastating results can be mitigated with the help of model SOC‐RDRO.

Benefit of directly solving model BW‐R

In this section, we solve the same instances through model BW‐R and compare the results with those obtained by solving model SOC‐RDRO.

First, we compare the in‐sample and out‐of‐sample metrics to show the stable convergence of model BW‐R. As the comparison results are similar, we exemplify them only by

$N=30$

in Figure 6. Figure 6a is similar to Figure 5a, with the exception that the metrics are plotted against c. For convenience of comparison, we duplicate Figure 5a in Figure 6b. As observed in the figures, solving model BW‐R achieves the same advantages as solving model SOC‐RDRO and avoids deterioration. This is because a large R can result in a meaningless model with all solutions having the objective value of 0 for the other. In contrast, model BW‐R stably converges to BW maximization as c increases.

FIGURE 6

Comparison of in‐sample and out‐of‐sample SOC (left axis, solid line, and blue‐shaded region) and BW (right axis, dashed line, and green‐shaded area) between models BW‐R and SOC‐RDRO with

$N=30$

: (a) model BW‐R and (b) model SOC‐DRO

Second, we show the computational efficiency improved by model BW‐R. The comparison is implemented for both solving the model with off‐the‐shelf solvers (exemplified by Cplex 12.8.0) and enumeration‐based algorithm (exemplified by a branch‐and‐bound algorithm modified from Camm et al., 2006, the detail of which is shown in the Supporting Information, EC.2.2). The computation is terminated with a time limit of 3600 s.

Figure 7 presents the average computational time of solving model SOC‐RDRO with different R (solid line) and BW‐R with different c (dashed lines), where the color indicates different sample sizes. As shown in the figure, the computational time of solving model BW‐R with both Cplex and the branch‐and‐bound algorithm is significantly less than that of solving model SOC‐RDRO. For example, the computational time of solving model SOC‐RDRO with

$N=30$

by Cplex reaches the time limit while that of solving model BW‐R is less than 1 s. Furthermore, we find that solving the models with Cplex becomes very time‐consuming as R increases (c decreases). Therefore, we recommend that with a controllable number of alternatives (as in this example), enumeration‐based algorithms are more efficient than directly solving the models with Cplex.

FIGURE 7

Computational time of Cplex (

$N=10,30$

) and a branch‐and‐bound algorithm (

$N=10,30,300,3000$

): (a) average computational time of Cplex and (b) average computational time of branch and bound

Compared to previous robust models

To our best knowledge, Wang and Curry (2012) and Wang and Gutierrez (2022) are the only two studies that propose robust models of the SOC problem. Both models proposed by Wang and Curry (2012) and Wang and Gutierrez (2022) can be reformulated to the same max–min framework shown in formula (6) with different uncertainty sets as follows.

The models proposed in Wang and Curry (2012) can be reformulated as a max–min model with an event‐wise uncertainty set (Chen et al., 2020) as follows:

$${\mathcal{D}}_1=\left\{\mathbb{P}\left|(\stackrel{\sim }{\mathbf{u}},\stackrel{\sim }{h},\stackrel{\sim }{n})\sim \mathbb{P},\mathbb{P}((\stackrel{\sim }{\mathit{u}},\stackrel{\sim }{h})\in {\mathcal{U}}_n|\stackrel{\sim }{n}=n)=1,\mathbb{P}(\stackrel{\sim }{n}=n)=\frac{1}{N},\forall \stackrel{\sim }{n}\in [N]\right.\right\},$$

21

where

${\mathcal{U}}_n=\{(\stackrel{\sim }{\mathit{u}},\stackrel{\sim }{h})=\{{\stackrel{\sim }{u}}_1,\text{…},{\stackrel{\sim }{u}}_{|\mathcal{L}|},\stackrel{\sim }{h}\}|{\widehat{u}}_{nl}-{\overline{u}}_{nl}{z}_{nl}\le {\stackrel{\sim }{u}}_l\le {\widehat{u}}_{nl}+{\overline{u}}_{nl}{z}_{nl},{\sum }_{l\in \mathcal{L}}{z}_{nl}=T,0\le z_{nl}\le 1,\stackrel{\sim }{h}={\widehat{h}}_n,\forall l\in \mathcal{L}\}$

. The two parameters T and

${\overline{u}}_{nl},\forall n\in [N],l\in \mathcal{L}$

mean that for sample customer n, partworth utility of at most T levels are subject to uncertainty and vary within the range

$[{\widehat{u}}_{nl}-{\overline{u}}_{nl},{\widehat{u}}_{nl}+{\overline{u}}_{nl}]$

. The extent of the robustness of their model is determined by J and

${\overline{u}}_{nl}$

. Through numerical experiments, we find that the results for the model are similar for different T. Therefore, we fix

$T=2$

and only adjust

${\overline{u}}_{nl}$

to obtain the model with a different extent of robustness. As recommended by Wang and Curry (2012), the most frequently used method to specify

${\overline{u}}_{nl}$

is constructing it to a certain percentage of

${\widehat{u}}_{nl}$

. Here, we adjust the

${\overline{u}}_{nl}$

from 0 to 15% of

${\widehat{u}}_{nl}$

with step 0.1% to establish the models with different extent of robustness.

The model proposed in Wang and Gutierrez (2022) is a max–min model with a scenario‐based uncertainty set; that is, the set contains a finite number of possible distributions. Specifically,

$${\mathcal{D}}_2=\left\{{\mathbb{P}}_1,{\mathbb{P}}_2,\text{…},{\mathbb{P}}_I\right\},$$

22

where I is the number of possible distributions. The extent of the robustness of their model increases as more distributions are added to the uncertainty set. We increase the number of I up to 100 with a step of 2 to get the models with different extents of robustness. In the process, we add

${\mathbb{P}}_i,i\in [I]$

by drawing new samples from the oracle distribution.

We resolve the same instances in previous sections with

$N=30$

by the models proposed by Wang and Curry (2012) and Wang and Gutierrez (2022). We plot the metrics in Figure 8 the same way as Figure 5. As shown in the figures, all the models follow the same tendency, that is, a higher extent of robustness for the SOC problem corresponds to a higher BW. This indicates the generality of this finding. As such, a general principle for the robust product design with SOC as the objective is to find a product with a higher BW.

FIGURE 8

Comparison between

$y_n$

in model SOC‐D and

$s_n$

model SOC‐RDRO with

$N=30$

: (a) Wang and Curry (2012), (b) Wang and Gutierrez (2022), (c) model BW‐R, and (d) weighted model

The generality of the finding inspires us to apply a heuristic way to achieve robustness, that is, a weighted model of SOC and BW maximization, which is shown as follows:

23

where η is a parameter that shows the relative importance of BW in the optimization. The results of the weighted model are shown in Figure 8d. As shown in the figure, when put more weight on BW, more robust solutions for the SOC problem are obtained.

Case study

In this section, we validate the effectiveness of the proposed model through an empirical dataset. The case is a CA for a credit card design. Table 5 presents the attributes and candidate levels considered in the CA. A total of 946 customers were sampled, and each of them was presented with 13–17 paired comparisons between two credit card designs. The detail of the response data is available in the R package bayesm. ⁵

TABLE 5

Attributes and levels for the credit card design

Attribute	Level	Attribute	Level
Interest rate	High (1)	Bank	A (12)
	Medium (2)		B (13)
	Low fixed (3)		(14)
	Medium variable (4)
		Rebate	Low (15)
Rewards	Program 1 (5)		Medium (16)
	Program 2 (6)		High (17)
	Program 3 (7)
	Program 4 (8)	Credit line	Low (18)
			High (19)
Annual fee	High (9)
	Medium (10)	Grace period	Short (20)
	Low (11)		Long (21)

A hierarchical Bayesian approach was applied to estimate the partworth utilities for each customer. The partworth utility of the first level of each attribute is regularized to 0, and the empirical distributions of the partworth utility of the rest levels are shown in Figure 9. We divide the 946 samples into training (the first 630 samples), validation (the next 100 samples), and test sets (the last 216 samples). Different sizes of samples are randomly drawn from the training set to establish models SOC‐D, and SOC‐RDRO with different R, and BW‐D. And then, we implement cross‐validation on the validation set to get a proper R for model SOC‐RDRO. Finally, we calculate the out‐of‐sample metrics of the models over the test set. We implement a comparison among the established models. A benchmark to evaluate the quality of the solutions is the maximal

${\text{SOC}}_{\text{os}}$

and

${\text{BW}}_{\text{os}}$

over the test sets, that is, the optimal objective values of models SOC‐D and BW‐D established by the samples in the test set. We denote the values as clairvoyant ones in the following.

FIGURE 9

Partworth utilities of the nonbenchmark levels

Figure 10 plots the out‐of‐sample metrics of models SOC‐D, SOC‐RDRO, BW‐D, and the clairvoyant ones. As shown in the figure, both models BW‐D and SOC‐RDRO improve

${\text{SOC}}_{\text{os}}$

and

${\text{BW}}_{\text{os}}$

on average compared to model SOC‐D and shrink the variation at the same time. Moreover, model SOC‐RDRO reaches the clairvoyant

${\text{SOC}}_{\text{os}}$

for most of the time, and model BW‐D has the most stable out‐of‐sample metrics. As shown in Figure 9, there are no significant outliers. Therefore, model BW‐D also performs well. It has a stabler

${\text{SOC}}_{\text{os}}$

and a higher

${\text{BW}}_{\text{os}}$

compared to model SOC‐RDRO.

FIGURE 10

Out‐of‐sample metrics of the credit line design case: (a)

${\text{SOC}}_{\text{os}}$

and (b)

${\text{BW}}_{\text{os}}$

CONCLUSIONS

This paper proposes two models for the product design problem. One is a DRO model of SOC maximization, and the other is a winsorized model for BW maximization. Our key finding is the equivalence between the two models when robust parameters are set properly. The equivalence motivates other main results of the paper. First, the equivalence explains the phenomenon we observed from the solutions presented in previous literature (Camm et al., 2006; Wang et al., 2009; Wang & Curry, 2012), that is, the solutions claimed to be more robust (or conservative) to the SOC problem have higher BW. Second, the equivalence leads to a further customer classification, which further separates the buyers identified by the classic models into two segments by a threshold of consumer surplus. The further classification enables characterizing probabilistic choice for the SOC problem and defining the outliers for the BW problem. Third, the equivalence indicates a more efficient approach to solving the DRO model for the SOC problem. Through extensive numerical experiments based on synthetic and real‐life data, we show that the proposed framework outperforms its classical counterpart.

Focal areas for future research can be summarized in three aspects. First, another important objective, profit, should be studied in the future because it includes a key trade‐off between price and SOC. As the paper finds the connection between SOC and BW from the perspective of robust optimization, we expect the relationship between the three objectives founded. Second, the effect of robustness on other choice models should also be investigated. Although first‐choice models are the most widely applied approach in product design, the effect of robustness on the probabilistic choice model or others that do not base on the utility theory, such as ranking‐based models (Bertsimas & Mišić, 2019), should be investigated. Third, the interaction between products in a product line should be considered. Although we extend our model to accommodate product‐line design in the Supporting Information (EC.1.2.1), the results do not examine the substitution and similarity of the products in the product line. Further study on this issue will provide a better understanding of the product‐line design problem.

Footnotes

ACKNOWLEDGMENTS

The authors gratefully thank the review teams for their constructive comments to improve the paper. The authors appreciate Professor Teo Chung Piaw for his help in revising the paper. This research is partially supported by the National Natural Science Foundation of China (grant nos. 72188101, 72171129, 71991462). Maoqi Liu and Changchun Liu are currently supported by National Research Foundation, Singapore, and A*STAR, under its RIE2020 Industry Alignment Fund – Industry Collaboration Projects (IAF‐ICP) grant call (grant no. I2001E0059) – SIA‐NUS Digital Aviation Corp Lab.

1

We provide a brief introduction of CA in the Supporting Information (EC.1.1) and refer the readers to Rao () for more details.

2

Detail introduction of SPSS can be found at https://www.ibm.com/sg‐en/products/spss‐statistics.

3

Detail introduction of STATA can be found at https://www.stata.com/.

4

Detail introduction of scipy package can be found at https://scipy.org/.

5

Go to https://cran.r‐project.org/web/packages/bayesm/index.html for the dataset and estimation tool.

ORCID iD

Maoqi Liu

Changchun Liu

Zhi‐Hai Zhang

Supplementary Material

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