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Bilevel Game-Theoretic Optimization for Product Adoption Maximization Incorporating Social Network Effects Feng Zhou, Roger J. Jiao, and Baiying Lei, Member, IEEE

Abstract—Viral product design involves sophisticated interactions between product portfolio planning and viral marketing. However, social network effects are mainly considered in marketing-related activities, and there is still limited investigation of the interplay between product design and viral marketing. In the context of social networks, it is important to jointly leverage both viral product attributes and viral influence attributes for product adoption maximization and product line performance optimization. In order to deal with the joint optimization problem, this paper presents a systematic formulation of a bilevel decision-making strategy for viral product design based on the Stackelberg game theory. The product adoption maximization problem with viral influence and product attributes is modeled as the leader and the product portfolio optimization problem with product attributes is modeled as the follower. The interaction and coupling of these two optimization problems are addressed with a coordinate-wise optimization strategy, in which adoption maximization is tackled with an improved greedy algorithm and a hybrid Taguchi genetic algorithm. A case study of Kindle Fire HD tablets demonstrates the feasibility and potential of the bilevel decision-making strategy for viral product design, which is advantageous over the existing viral marketing methods that only consider viral influence attributes. Index Terms—Bilevel optimization, product adoption maximization, social network effects, viral product design.

I. I NTRODUCTION OCIAL network research has been increasingly attracting the attention of scholars from the domains of marketing, computer science, sociology, and engineering [1].

S

Manuscript received January 29, 2015; revised April 26, 2015; accepted July 4, 2015. This work was supported in part by the National Natural Science Foundation of China under Grant 61402296 and in part by the National Natural Science Foundation of Guangdong Province under Grant S2013040014448. This paper was recommended by Associate Editor F. Wang. (Corresponding author: Baiying Lei.) F. Zhou was with the George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30322 USA. He is now with MediaScience, Austin, TX 78753 USA (e-mail: [email protected]). R. J. Jiao is with the George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30322 USA (e-mail: [email protected]). B. Lei is with the National-Regional Key Technology Engineering Laboratory for Medical Ultrasound, Guangdong Key Laboratory for Biomedical Measurements and Ultrasound Imaging, Department of Biomedical Engineering, School of Medicine, Shenzhen University, Shenzhen 518060, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSMC.2015.2476473

One of the significant reasons is that social networks have revolutionized the way that humans interact, such as that between marketers and customers and among customers themselves. Three important effects have been identified in social networks, including word-of-mouth effects, imitation effects, and network effects [2]. First, the word-of-mouth effects regarding a product, a service, or a brand often speed-up the process of information diffusion and awareness [3]. Second, increased adoption of specific products or services often leads to herding behavior, especially when there is homophily [4] in the social network and the number of adopters has surpassed the threshold [5]. This process is often depicted as the result of imitation effects [2]. Third, a network effect (or network externality) refers to the effect that one user of a product or a service has on the value of that product to other people [6]. When a product (e.g., a smartphone) is prone to a positive network effect, the more people adopt it in the social network, the more valuable it is to each customer. It thus increases the tendency of product adoption. In this paper, we refer to these three effects mentioned above as social network effects. The underlying factors that lead to such social network effects are the viral attributes either in the social network or in the product. Therefore, companies are increasingly proactive in making use of such effects by strategically identifying viral attributes in the social network in order to achieve their goals. In particular, one of the important problems is viral marketing, which can be understood as an optimization problem, i.e., influence maximization (InfMax) [7]. This is directly linked to viral influence attributes. Viral influence attributes are embedded in the social network rather than in the product. One of the most important types is a seed customer that corresponds directly to the viral mechanism of the product. Seed customers, offered incentives by firms, are able to influence the largest possible number of other customers to adopt a product (e.g., social hubs within a social network [8]). Thus, how to identify seed customers is important to the InfMax problem. Given a directed social network and a certain budget, the key is to find a seed set so that by activating them, we can maximize the expected number of customers that eventually get activated based on a diffusion model (e.g., linear threshold models and independent cascade models [9]). In order to effectively solve this problem, multiple algorithms are proposed. For example, a greedy approximation algorithm is first proposed that guarantees the influence spread is within 1 − 1/e of the optimal influence spread.

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Based on this algorithm, multiple improved algorithms have been proposed in terms of computational efficiency [10]–[12] and scalability [13]. However, one of the limitations of these methods is that they only consider the social network effects in the product diffusion and adoption process, without taking into account customers’ preferences to products, especially viral product attributes that can spread widely in a social network. For example, Aral and Walker [14] acknowledged the effects of social contagion on product planning and argued that certain viral product attributes can be engineered before launching the product. Viral product attributes are fundamentally about the content of a product and the psychological effects that can have on a customer’s desire to share the product with peers [15], such as the attractive attributes in the Kano model [16] that delight customers unexpectedly and satisfy customers’ latent needs. One good example is probably the vanity mirror mounted on the sun visor in the car. When it was first introduced, it was among the most popular dealeradded accessories that provided high profit margins with the sales [17]. It became viral among women and then among all the passengers. However, as time goes by, it becomes a basic product attribute. Another stream of work in engineering design and product portfolio planning (or product line design) aims to optimize product performance for profit-driven goals. For example, decision-based design considers customer heterogeneous preferences in the decision-making process and formulates an enterprise-level goal to maximize the utility of a company [18]. The share-of-choice (SoC) problem is another important research task [19]. Given a set of attributes A, the objective is to identify J product configurations derived from K product attributes with appropriate attribute levels, so that the expected number of adopters can be maximized. The SoC problem shares the same objective with the InfMax problem. Despite the fact that the SoC problem considers the customer preferences in terms of product attributes, it ignores the interactive relations between customers. Given these two questions, it is plausible to combine these two critical aspects for viral product design, which is referred to as adoption maximization with viral attributes (AdpMaxVA) [20]. However, the SoC problem involves a combinatorial optimization of attribute levels, which are subject to engineering constraints and goals, such as customer satisfaction and costs. These engineering concerns must be taken into consideration in the design and manufacturing process in order to maximize product profit. Therefore, the AdpMaxVA problem involves a bilevel decision-making process, in which the upper-level optimization model aims to maximize product adoption, whereas the lower-level optimization model aims to maximize product line performance in terms of costs and customer satisfaction. Such a design paradigm jointly solves these two problems by bridging the gap between the marketing and the engineering domains. Toward this end, the main contributions are summarized below. 1) We formulate viral product design as a bilevel game theoretic optimization problem. The objective of viral

product design is to select an optimal set of viral product attributes for product portfolio planning, and to find an optimal set of viral influence attributes, specifically an optimal set of seed customers, so as to jointly maximize the expected number of product adopters. Such a formulation implies a new paradigm of design problems by incorporating the social aspect, especially social network effects. The interplay of product design and social network effects necessitates an integration of these two optimization problems within a coherent framework. 2) We propose a coordinate-wise optimization strategy to tackle the bilevel game-theoretic optimization problem by fixing one type of variables (i.e., seed set or product attribute) at a time in the optimization problem. When the product attributes are fixed, it becomes an InfMax problem, which is solved by the cost-effective lazy forward (CELF++ ) algorithm [12]. When the seed set is fixed, the bilevel optimization problem is first transformed into a single-level parametric optimization problem, which is solved by a hybrid Taguchi-genetic algorithm (HTGA) [21]. 3) We conduct a case study with a real-world semantically rich data set from Amazon.com to demonstrate the potential and feasibility of the proposed method. The social network is constructed based on the reviewer– commenter links about Kindle Fire HD tablets, and the candidate sets of product attributes and attribute levels are derived based on the review content. II. R ELATED W ORK A. Influence Maximization In viral product design, InfMax is one of the most frequently studied problems. Two important aspects of InfMax are identified, including diffusion modeling and algorithm development. Two classic types of diffusion models, including independent cascade models and linear threshold models [9], will be discussed here. Given a social network G modeled as a directed graph, each node is either active (i.e., product adopter) or inactive. Each node’s tendency to become active increases monotonically as more of its neighbors become active. The adoption process unfolds in discrete steps. Assuming v is an inactive social entity (i.e., nonadopter) at step t, and the probability of v becoming active (i.e., adopters) depends on the influence of v’s neighbors who are adopters at step t. In the linear threshold model, each node v has a threshold ϑv uniformly distributed between 0 and 1. When the joint influence by each neighbor u surpasses the threshold ϑv , the node v will become active, i.e., u∈N a bu,v ≥ ϑv , where N a is the set of active neighbors of v, and bu,v is the influence from u to v. In independent cascade models, the activation probability is calculated as pv = 1 − u∈N a (1 − pv,u ) [10], [13], where pv,u is the influence probability of u on v. It can be seen that the adoption probability is calculated based on the assumption that each active neighbor influences v independently. Besides, pv,u is often set as 1/k for all u ∈ N a , where k is the total number of active neighbors of v [10], [13], or a constant,

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such as 0.1 or 0.01 [9], [22]. Such a simple model cannot accommodate the dynamic nature of the adoption process in online social networks, and thus further investigation is needed. Based on these two classic models, extension has been proposed. In order to capture product information in the social network, Bhagat et al. [7] proposed a linear threshold model with colors, in which both influence of neighbors and product evaluations are incorporated to predict product adoption. Similarly, Chen et al. [23] proposed a diffusion model based on the independent cascade model, and incorporated negative bias by dividing the active state into two substates, i.e., positive and negative. In terms of algorithm development, Kempe et al. [9] proved that the InfMax problem is NP-hard, and proposed a greedy approximation algorithm within 1 − 1/e of the optimal influence spread. The performance is significantly better than classic degree and centrality-based heuristics in influence spread. However, this algorithm is still computationally intensive as it involves a sufficient number of Monte Carlo simulations to estimate the influence spread. In order to improve its efficiency, Leskovec et al. [11] proposed a CELF method that exploits the submodularity property of the InfMax problem. Mathematically, function σ is submodular iff (S∪{v})−σ (S) ≥ σ (T ∪{v}) − σ (T) whenever S ⊆ T. This property significantly reduces the number of evaluations on the influence spread. Chen et al. [10] proposed fine-tuned heuristics to improve the original greedy algorithm and the CELF method. Experimental results show higher efficiency. Subsequently, Chen et al. [13] designed a new heuristics algorithm that is scalable to large social networks (with millions of nodes and edges), controlling the balance between efficiency and InfMax. Goyal et al. [12] improved CELF by further exploiting submodularity and proposed CELF++ , which improves the efficiency of CELF by 35%–55%.

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The bilevel optimization problem is originated from the field of game theory with a hierarchical optimization problem [25]. It is a special kind of optimization in which a lower-level optimization problem (the follower) is nested within an upper-level optimization problem (the leader) [26]. In such a structure, the leader and the follower compete against each other so that the leader makes a decision first and then the follower reacts optimally to the leader’s decision as feedback [27]. Based on the follower’s decision, the leader adjusts accordingly. The process stops when both obtain satisfactory solutions. However, the solution to the bilevel programming is often hard to obtain, mainly due to the nonconvexity [28]. Even a linear–linear bilevel programming problem is NP-hard [29]. Traditional solutions include vertex enumeration, replacing the lower-level problem with its Karush–Kuhn–Tucker (KKT) conditions (when it is convex and continuously differentiable) and applying gradient methods [28]. However, these approaches are not computationally efficient, especially for large problems, and sometimes lead to a paradox that the follower’s decision power dominates the leader’s [30]. A better way to solve this is to apply evolutionary algorithms, such as genetic algorithms (GAs), to dealing with complex optimization problems [31]. For example, Calvete et al. [28] combined classical enumeration techniques and GA to achieve near-optimal solutions within acceptable computational time. Ji et al. [32] tackled a leader–follower joint optimization problem of technical system modularity and material reuse modularity with a constrained GA. The advantage of these methods is that GAs usually have a low risk of ending up with a local optimum and are not constrained by the assumptions (e.g., KKT conditions) in the classical methods. III. B ILEVEL O PTIMIZATION FOR V IRAL P RODUCT D ESIGN A. Bilevel Optimization

B. Bilevel Optimization The prevailing methods address the InfMax problem as a single-level optimization problem. For example, Barbieri and Bonchi [20] integrated viral product attributes and social network effects in order to maximize product adoption as a single-level optimization, without considering the engineering concerns regarding product attributes. This is also the typical practice in decision-based design, in which enterprise-level goals and engineering-level design decisions are integrated as a single-level utility objective [18]. However, viral product design involves both viral product attributes and viral influence attributes. The former implies a product portfolio-planning problem while the latter implies an InfMax problem. These two problems usually have a certain degree of autonomy and may involve conflicting decisions as well. While multiobjective optimization approaches can well accommodate standalone design problems, they often fail to capture the interaction and coupling involved in the optimization process of product portfolio planning and InfMax. As pointed out by Deb and Sinha [24], the bilevel optimization strategy is fundamental to such joint optimization problems, which can deal with the interaction and coupling for conflicting decisions.

Generally speaking, the bilevel formulation can be represented as follows [26], [33]: MaxX s.t.

F(X, Z) G(X, Z) ≤ 0, X ∈ Rs

(1.1) (1.2)

f (X, Z) g(X, Z) ≤ 0, Y ∈ Rt

(1.3) (1.4)

where Z solves MaxZ s.t.

where G and g are vector-valued functions of dimensions p and q, showing the constraints. In (1), we denote the constraint region as = {(X, Z) : G(X, Z) ≤ 0, g(X, Z) ≤ 0, X ∈ Rs , Z ∈ Rt }. The projection of onto the upper-level design space gives the feasible set for X, i.e., U = {X ∈ Rs : ∃Z ∈ Rt , such that (X, Z) ∈ }. Then the follower’s rational reaction set for X ∈ U can be defined as R(X) = {Z ∈ Rt : Z ∈ ¯ : g(X, Z) ¯ ≤ 0}}. The most important concept arg min{f (X, Z) is the inducible region, which defines the feasible set at least when the lower-level optimization model has a unique optimal solution for all values of X, and it can be defined as IR = {(X, Z) : (X, Z) ∈ , Z ∈ R(X)}.


Fig. 1.


System model of bilevel decision-making for viral product design.

We assume that the bilevel optimization problem is well posed in that is nonempty bounded, and IR is nonempty. In addition, we assume that R(X) is single-valued, which implies that there exists a unique response function, Z = Z (X). With these assumptions, we can find an optimal solution to (1), denoted as (X∗ , Z∗ ) based on the joint optimization of product adoption and product portfolio planning. Solving the bilevel model can be realized in three steps. Step 1: The leader makes a decision, X , according to the leader’s own strategy F(X, Z); and then announces the decision to the follower with a set of constraints. Step 2: The follower(s) makes a decision subject to its own strategy f together with the leader’s decision; and then feedback the follower’s feasible solution, Z = Z(X ), to the leader. Step 3: The leader adjusts its decision to obtain a new X , based on the follower’s feasible solution. These steps are iterated until a satisfactory result is obtained for both the leader and the follower(s) [32]. However, such solution is often not efficient. Based on the unique response function Z = Z (X), (1) can be converted into a single-level parametric optimization problem [26] as follows: Max F X, Z (X) (2.1) s s.t. G X, Z (X) ≤ 0, X ∈ R . (2.2) Equation (2) is essentially a bilevel optimization problem, and it paves the way for solving such problems without directly going through the three steps. B. Formulating Viral Product Design Problem Viral product design involves a leader–follower joint optimization problem, that is, to optimize product portfolio (i.e., product line) planning and maximize product adoption jointly. It leverages both marketing and engineering concerns as shown in Fig. 1. In this paper, the leader F leverages both viral product attributes and social network effects for product adoption maximization as the marketing goal. The basic idea is to identify an optimal set of attributes and attribute levels and an optimal set of seed customers that can maximize product adoption in the social network. Thus, the leader F has a s-dimensional design variables X ∈ Rs . The follower f has

a t-dimensional design variables Z ∈ Rt , including attribute levels and product variants. The follower considers both customer satisfaction and engineering cost that can maximize the expected overall performance of the entire product line as the engineering goal. Between the leader and the follower, the social network is where the decision is tested. A leader–follower model assumes certain decision power for both the leader and the follower, with the leader possessing a higher priority. 1) Upper-Level Optimization Model: a) SoC problem: Viral product design involves selecting an optimal set of viral product attributes and attribute levels. This is related to product portfolio planning, which aims at the selection of a near-optimal mix of product variants configured by different product attribute levels to offer in the target market [34]. A product line involves multiple product variants (e.g., 16 GB, 32 GB, and 64 GB models of Kindle Fire HD tablets), which attract heterogeneous customers in the market. Each product variant is configured with K product attributes, i.e., ak ∈ A, k = 1, . . . , K. Let Lk be the number of levels for the kth attribute. Then the attribute level set can be represented as A∗ = {a∗kl |k = 1, . . . , K, l = 1, . . . , Lk }. Considering all the possible configurations, there are a number of meaningful J product configurations, indicated by X = {Xj |j = 1, . . . , J}, and Xj = (Xjk |k = 1, . . . , K) = (xj11 , . . . , xj1L1 , . . . , xjK1 , . . . , xjKLK ) is a vector showing a particular configuration for the jth product Pj , where xjkl |k = 1, . . . , K, l = 1, . . . , Lk is 1 if the lth attribute level of the kth attribute is selected for Pj , otherwise it is 0. Y = (yj |j = 1, . . . , J) is a vector indicating a particular choice of Pj , and yj = 1 if Pj is chosen, otherwise it is 0. The basic principle of identifying an optimal set of product attributes and attribute levels is the SoC problem [19]. For a set of customers, V = {v1 , v2 , . . . , vN }, they have different part-worth utilities, i.e., uikl , which denotes the utility of the lth attribute level of kth attribute perceived by the ith customer in the social network. The SoC adoption model postulates that customer vi adopts the jth product only if his holistic utility of the product Uij exceeds his hurdle utility, hij . In this situation, zij = 1, otherwise it is 0. The objective of the SoC problem is N to maximize the number of product adopters, i.e., i=1 zi = N J z , denoted as σ (X). Therefore, the SoC problem i=1 j=1 ij can be formulated as follows [27], [35]: Max

σ (X) =

N

zi =

i=1

s.t.

Uij =

Lk K

N J

zij

(3.1)

i=1 j=1

uikl xjkl ≥ hij zij ,

k=1 l=1 Lk

xjkl = l=1 Lk K

∀i ∈ [1, N], j ∈ [1, J]

(3.2)

1, k ∈ [1, K], j ∈ [1, J]

(3.3)

xjkl − xj kl > 0, j = j

k=1 l=1 J

yj ≤ J + < J

j=1

(3.4)

(3.5)


zi =

J

zij , j ∈ [1, J]

(3.6)

j=1

zij ∈ {0, 1}, xjkl ∈ {0, 1}, ∀i ∈ [1, N], j ∈ [1, J], (3.7) k ∈ [1, K], l ∈ [1, Lk ]. The exclusive constraint (3.3) guarantees that only one attribute level is selected for each product attribute. Constraint (3.4) denotes that at least one attribute level is different for two different product variants. J + in constraint (3.5) is the upper bound for the total number of product variants offered in the market. b) AdpMaxVA problem: Viral product design also necessitates an AdpMaxVA problem, that is, how to select a set of key customers to act as the seeds to influence the largest number of customers so that they are likely to become product adopters. Given a directed social network G = (V, E), where V = {v1 , v2 , . . . , vN } is a set of nodes, indicating a set of customers and a link from vj to vi , i.e., (vj , vi ) ∈ E means that vj can potentially influence vi . Given a certain budget, the key is to find a seed set S ⊆ V with |S| = n < N, so that by activating them, we can maximize the expected number [denoted as σ (S)] of customers that will eventually get activated based on a diffusion model. Assume Iit is the influence of social network effects of vi from his or her active neighbors at time t. Iit can be calculated by aggregating individual influence from active neighbor vj to vi , i.e., Ijit at time t. We introduce the adoption probability by integrating the InfMax and SoC problems [20] Pr Pj |i, t =

exp ϕit Pj 1 + exp ϕit Pj

2) Lower-Level Optimization Model: For product portfolio planning, it is necessary to find an optimal combination of product attributes and attribute levels for best product performance in terms of utility-to-cost ratio. In this aspect, the lower level model considers both customer satisfaction and engineering cost so that the expected overall performance of the entire product line can be maximized. Among many, a shared surplus measure is selected. It leverages both customer satisfaction and enterprise profitability formulated as [36] f (Z) = φ(X, Y) =

N J Uij

Cj

i=1 j=1

zi yj

(7)

where Uij is the utility of the ith market segment (or ith customer) for the jth product. It is normally the case that the market can be grouped into several segments, in which all the customers are homogeneous and their utilities are similar. Uij (9.11) has a copulas formulation of part-worth utilities derived from cumulative prospect theory of the attribute levels of product Pj [37]. Cj (9.12) is a cost function of product Pj . We model the cost consequences of providing variety based on variation of process capabilities [38]. Finally, by maximizing f (Z), we can obtain the set of product attributes and their levels, which can be represented as Z∗ = arg max[φ(Z)] = Z|∀Z : φ Z ≤ φ(Z) . (8) C. Bilevel Optimization Model

(4)

where ϕit

5

Pj = Iit + Uij − hij

(5)

By compiling the upper-level optimization model and the lower-level optimization model, the bilevel optimization model is computed as follows: Max

F(X, Z) = σ (S, X, Y) =

N J

(9.1)

zij

i=1 j=1

and Pr(Pj |i, t) is the probability that vi will adopt product Pj at time t + 1 if it is not smaller than the adoption threshold ϑi at time t. In this case, zij = 1 if the product is Pj , otherwise it is 0. Here, ϕit (Pj ) consists of social network influence at time t, i.e., Iit , customer i’s preference to product j, i.e., Uij (7), and customer i’s hurdle utility of product j, i.e., hij . This formulation enables both viral product attributes and viral influence attributes to be incorporated in the product adoption model, and therefore, it is referred to as AdpMaxVA, which becomes the upper-level optimization model, that is

Max F(X, Z) = σ (S, X, Y) =

N i=1

zi =

N J

S ⊆ V.

(9.2) (9.3)

j=1

yj , ∈ {0, 1}, j ∈ [1, J] Lk

xjkl = l=1 Lk K

(9.4)

1, k ∈ [1, K], j ∈ [1, J]

(9.5)

xjkl − xj kl > 0, j = j

zij

(6.1)

i=1 j=1

s.t. Pr Pj |i, t ≥ ϑi , ∀i ∈ [1, N], j ∈ [1, J] |S| = n

s.t.

sgn Pr Pj |i, t − ϑi + 1 zij = 2 ∀i ∈ [1, N], j ∈ [1, J] J yj ≤ J +

xjkl ∈ {0, 1}, ∀i ∈ [1, N], j ∈ [1, J] k ∈ [1, K], l ∈ [1, Lk ]

(6.2) (6.3) (6.4)

Here the design variables are the seed customers [i.e., S, corresponding to X in (1.1)] and attribute levels [i.e., X and Y corresponding to Z in (1.1)].

(9.6)

k=1 l=1

|S| = n S⊆V

⎡

Z∗ = arg max⎣f (Z) = φ(X, Y) =

J N Uij i=1 j=1

Cj

(9.7) (9.8) (9.9) ⎤ zi y j ⎦

(9.10)



s.t.

Uij = c1 ϕ

−1

K

ϕ ξk + (1 − ξk )

k=1

Lk

xjkl uikl

l=1

+ c2 + εij 3σjT 1 = exp Cj = exp PCIj μTj − LSLT

(9.11)

(9.12) zi =

J

zij , j ∈ [1, J]

(9.13)

j=1

gjk (X) ≥ 0, j ∈ [1, J], k ∈ [1, K].

(9.14)

In this formulation, (9.1)–(9.9) form the upper-level optimization model while (9.10)–(9.14) form the lower-level optimization model. Equation (9.2) denotes the adoption condition, that is, if the adoption probability Pr(Pj |i, t) is larger than ith customer’s adoption threshold ϑi , zij = 1, otherwise zij = 0. Sgn is the sign function. Equation (9.14) describes the technical requirements and constraints of product j for the k attribute domains. The leader controls the follower through product attributes X and product variants Y, indicating a priority for determining the product configuration. The follower, on the other hand, attempts to optimize a shared surplus φ(X, Y) between the customers and the producer subject to the decisions made by the leader. The follower returns an optimal configuration X ∗ and selected product variants Y to the leader, which influences Uij in the upper-level, and in turn zij in the upper-level’s optimization model. Therefore, the bilevel optimization captures such interactions and coupling between the engineering domain and the marketing domain. The optimal solution (S∗ , X ∗ , Y ∗ ) thus indicates the “best compromise” between these two domains, where S∗ is the final set of customer seeds and X ∗ is the set of attribute levels that configure the product variants Y ∗ in the product line. IV. M ODEL S OLUTION A. Strategy for Solution The formulation in (9) is an integer nonlinear bilevel programming problem. It actually involves two different kinds of variables, i.e., seed customers (S) and product attributes and variants (X, Y). In order to deal with this, we adopt a coordinate-wise optimization strategy based on [20] by decomposing it into two subproblems and alternate the procedure of greedy seed selection and attribute update in the iterations of the search process. The greedy seed selection is solved with CELF++ [12] due to its high efficiency, while the attribute update problem is solved by an HTGA [21], [39] method. For the ith iteration, improvement of the objective function σ (S, X, Y) can be obtained, by forcing one of the following inequalities to hold: (10.1) σ Si , X i , Y i ≤ σ Si+1 , X i , Y i i i i σ S , X , Y ≤ σ Si , X i+1 , Y i+1 . (10.2) When the product attributes and their levels are fixed, the greedy seed selection algorithm is applied to increase the size

Fig. 2.

Solution procedure for the bilevel optimization model.

of the seed set S by making (10.1) hold. When the seed set is fixed, the HTGA method is utilized to update the attribute levels and their corresponding product variants. It is observed that the seed set S does not directly relate to lower-level optimization problem. However, when the seed set S is fixed, both X and Y interact between the upper-level model and the lowerlevel model. Furthermore, σ is only a function of X and Y. By making use of the single-level parametric optimization problem introduced in (2), we can transform the bilevel optimization problem in (9) into a single-level form similar to (2), which is shown as follows: Max s.t.

F(X, (X)) = σ (X, (X))

G (X, (X)) ≥ 0, X ∈ {0, 1}

(11.1) s

(11.2)

where Y is a unique response function of X, i.e., Y = (X). Then (11) is solved with the proposed HTGA method. Therefore, the overall solution procedure is illustrated in Fig. 2. Due to the fact that the InfMax problem has been well addressed in the previous studies [10], [12], we mainly introduce the HTGA method that solves the bilevel optimization problem. B. Hybrid Taguchi-Genetic Algorithm The HTGA method is based on orthogonal GAs, in which some major steps (e.g., crossover) of a GA algorithm can be considered as “experiments” and the Taguchi’s method or orthogonal design can be used to identify the global optimum more robust and statistically sound [40]. The Taguchi’s method [41] can produce orthogonal arrays, i.e., fractional factorial design, which can scan the feasible space more evenly to locate good points for further exploration in the following iterations [42]. Tsai et al. [21] introduced the notion of signalto-noise ratio (SNR) to select better offspring, which further


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TABLE I O RTHOGONAL A RRAY FOR 12 P RODUCT ATTRIBUTES OF K INDLE F IRE HD TABLETS

TABLE II P RODUCT ATTRIBUTES AND ATTRIBUTE L EVELS I DENTIFIED FOR V IRAL P RODUCT D ESIGN

increases the quality and robustness of a GA, termed as the HTGA. In this paper, we adopt their idea and extend the original two-level orthogonal arrays to n-level ones in the context of a bilevel optimization problem. 1) Taguchi Method: Both orthogonal arrays and SNR are used in the Taguchi method to generate high quality chromosomes in HTGA. Each product attribute is considered as a factor in the experiment design process, and each attribute level is corresponding to a factor level. Usually multiple levels are associated with a factor. Therefore, we extend the original two-level orthogonal arrays in [21] to n levels. An orthogonal array is a fractional factorial matrix with balanced comparison of factor levels and their interactions [43]. For K factors with q levels per factor (when the numbers of factor levels are different, dummy variables can be used), the general symbol for a standard orthogonal array can be represented as (12) Lm qm−1 = ai,j m×(m−1) where L denotes a Latin square, m = qk is the number of experiment runs (i.e., number of rows in the orthogonal array), k > 1 is a positive integer, and m−1 is the number of columns in the orthogonal array. Thus standard orthogonal arrays can be represented as L4 (23 ) or L27 (326 ). They can be generated by algorithms [42] or software (e.g., SPSS). Table I shows part of Lm (qm−1 ) = L27 (326 ) generated by SPSS, which is used in the case study. It has K = 12 factors (i.e., attributes) and each has two or three levels as shown in Table II. Since K ≤ m − 1, only the first K = 12 columns are used, while the other m − 1 − K = 14 columns are ignored.

The notion of SNR is introduced to the Taguchi method in order to improve the robustness of the design. For example, turning the ignition key (signal factor) should always start the car, regardless of different noise factors, such as temperature, gasoline type, and engine wear. Three types of SNR tests exist based on the direction of the optimal path of the response variable [43]: 1) nominal-is-best; 2) smaller-the-better; and 3) larger-the-better. In this paper, the larger-the-better case is adopted. Given a sample of T observations, {oi |i = 1, . . . , T}, its SNR is calculated as follows: T 1 1 SNR = −10log . (13) T o2 i=1 i 2) HTGA Process: The HTGA process is shown in Fig. 2. Before we introduce the HTGA process, the chromosome coding and the definition of the fitness function are described. a) Chromosome coding: We code a chromosome with only one product configuration at a time, i.e., Xj . Based on Table II, we transform the binary string of Xj into Xj = {xj1 , xj2 , . . . , xjK }, where xjk ∈ {1, 2, 3}, representing the specific level chosen by Pj . This operation has several merits. First, this dramatically reduces the length of the binary string, which increases the efficiency of the HTGA process. Second, the best J + product configurations can be selected based on the seed set S and attribute level set A∗ , and these product configurations determine the value of Y. Therefore, a chromosome cj of an upper-level decision variable Xj can be represented as a string, i.e., cj = Xj = {xj1 , xj2 , . . . , xjK }, where xjk ∈ {1, 2, 3} (see Table II).



TABLE III O PTIMAL C HROMOSOME G ENERATION P ROCESS BASED ON THE H YBRID TAGUCHI O PERATION

b) Fitness function: We propose to apply an exact penalty function to handle constraints involved in (9) [39], which can be represented as follows: σ cj , cj + p cj , cj , cj exists (14) fit cj = −∞, otherwise

where p(cj , (cj )) = Cp min(G (cj , (cj )), 0), Cp is the penalty constant and takes a large positive value. It shows that when G (cj , (cj )) < 0, p(cj , (cj )) will have a large negative value, which substantially reduces the value of fitness function. c) Initialization: We generate a random chromosome cj = Xj = {xj1 , xj2 , . . . , xjK }, in which xjk randomly take values of 1, 2 or 3. This process is repeated for Mp times, which forms the initial population with Mp randomly generated chromosomes. d) Selection: In the first iteration, we will retain all the Mp chromosomes. However, in the following iterations, we will have only retain the top Mp chromosomes by evaluating the fitness function. This is because more chromosomes will be generated with the following process of HTGA. e) Crossover: It is a genetic operator that can vary the programming of a chromosome from one generation to the next. In this paper, a simple but effective one-point crossover technique is used with a crossover probability pc . For two randomly selected parents, c1 = {x11 , . . . , x1i , x1,i+1 , . . . , x1K } and c2 = {x21 , . . . , x2i , x2,i+1 , . . . , x2K }, it randomly selects the ith cut-point, and exchanges the right parts of them to form two new generations, c1 = {x11 , . .. , x1i , x2,i+1 , . . . , x2K } and c2 = x21 , . . . , x2i , x1,i+1 , . . . , x1K . This process is repeated for Mp times. f) Hybrid Taguchi operation: In order to illustrate the process of the hybrid Taguchi operation, we use the example in Table I. An orthogonal array L27 (326 ) is produced with only the first 12 columns, in which each factor has two or three levels as shown in Table II. Each row forms a configuration of the product, Xj . For example, for the first factor connectivity, it has three levels (L1 = 3), WiFi, 4G, and both, indicated as “1,” “2,” and “3” in Table I. If WiFi is selected in the first row,

then x11 = 1. For the purpose of illustration, we identify an optimal product configuration that can maximize a simplified version of the shared surplus in φ=

12 uk πk

(15)

k=1

where the part-worth utility uk is derived via a customer preference model based on cumulative prospect theory [37] and its cost πk is derived with (9.12). Their values are shown in Table III. We assume that three chromosomes are selected to determine which level is the best one for each factor, i.e., c1 = {2, 1, 3, 1, 3, 1, 2, 1, 1, 2, 3, 1}, c2 = {3, 2, 1, 3, 2, 2, 1, 2, 2, 1, 1, 2}, and c3 = {1, 3, 2, 2, 1, 3, 3, 3, 3, 3, 2, 3}. Then the values in c1 , c2 , and c3 are considered as the new level 1, level 2, and level 3, respectively. For example, for battery, level 1, level 2, and level 3 are now 3, 1, and 2, respectively. By filling in Table I with these new levels, we can calculate their function values using φ in (15) from the first experiment to the 27th experiment. Define the effects of factors as Efl = sum of SNRi for factor f at level l

(16)

where SNRi is the SNR for the ith experiment. In Table I, we first calculate SNRi , i = 1, . . . , 27, for the 27 experiments according to (13). Then (16) is used to compute Efl , f = 1, . . . , 12, l = 1, 2, 3. By comparing the values among Ef 1 , Ef 2 , and Ef 3 , the largest value corresponds to the optimal level based on the larger-the-better criterion (see Table III). Note that in the optimal level row, c2 (1) denotes the first gene of the chromosome c2 , which is 3, i.e., the first gene of the produced chromosome. Thus, the final produced chromosome is cp = {3, 1, 1, 1, 2, 3, 2, 2, 1, 2, 3, 1}. Using (15), we obtain their shared surplus of the four product configurations corresponding to the four chromosomes, φ(c1 ) = 8.32, φ(c2 ) = 10.75, φ(c3 ) = 5.38, and φ cp = 11.26. This validates that the produced chromosome is the optimal one among the four. Thus, this hybrid Taguchi operation can produce better offspring with three parents without computing all the possible product configurations (i.e., 36 × 26 = 46 656), but with 27 orthogonal


configurations only. This process greatly improves the convergence rate of the proposed HTGA method, especially important for the complex bilevel optimization problem in this paper. g) Mutation: It is a genetic operator that is used to maintain and introduce genetic diversity from one generation to the next. In this paper, we shall first randomly choose a chromosome from those generated in the previous steps with probability pm . Then two randomly selected genes from the chromosome will be altered from one level to another level (e.g., from level 1 to level 2 or level 3 with an equal probability). This process is repeated for Mp times. h) Fitness function evaluation: All the chromosomes will be evaluated in terms of their fitness using the fitness function in (14) to keep the top Mp chromosomes as the new population for the next iteration until the stopping criterion is met. i) Stopping criteria: It is controlled by a maximum iteration number Imax . However, when the iteration number is smaller than Imax , the convergence of the algorithm is also checked by comparing the optimal fitness function values for the most recent two iterations. If the optimal fitness function value does not change for over a minimum number of times Cmin , the algorithm would have converged and will be halted. V. C ASE S TUDY A. Data Collection A case study of Kindle Fire HD tablets (released in 2012) from Amazon.com is used to illustrate the performance of the proposed method. We crawl reviews of Kindle Fire HD tablets (both 7- and 8.9-inch models) from September 2012 to September 2013 (50 weeks), and the social network is constructed based on the reviewer–commenter links to show the possible influence from reviewers to commenters. A total number of 12 product attributes are identified by sentiment analysis as shown in Table II and each has two or three levels. Note that these product attributes and attribute levels are obtained based on our previous study [44], and they are related to both popularity and latent customer needs that often delight customers unexpectedly. We assume that latent customer needs, satisfied by certain attributes, possibly contribute to product adoption in the social network. The case study is to illustrate the proposed bilevel optimization method by identifying the optimal product configuration in terms of product adoption and shared surplus maximization under the influence of social network effects. B. Data Analysis The input of the bilevel optimization includes the following. 1) The social network constructed from the reviewer– commenter links about Kindle Fire HD tablets G = (V, E), where V = {v1 , v2 , . . . , vN }, N = 5220 is a set of customers and a link from vj to vi , i.e., (vj , vi ) ∈ E means that vj can potentially influence vi . The total number of links is 10 467. 2) A set of product attributes A = {a1 , . . . , aK }, K = 12 and a set of their attribute levels in Table II, i.e., A∗ = {a∗kl }, k = 1, . . . , 12, l = 1, . . . , Lk , Lk = 2 or 3.

9

3) The budget indicated as a set of seed customers S, |S| = n < 5220. We choose different values of n for experimentation. 4) The diffusion model in (6) and its parameters, including social network effects on customer vi , i.e., Iit , the utility of the ith market segment (or ith customer) for the jth product, Uij , and customer vi ’s hurdle utility for the jth product, hij , indicating equilibrium between adopting and not adopting the product. Iit is estimated using our previous method based on three underlying factors of the social network effects, including interaction strength, structural equivalence, and entity similarity [45]. In order to estimate Uij , we segment the market based on the mc customer characteristics from their profiles on Amazon.com. Using a data mining method named rough set [46], we identified five market segments based on the adoption categories, including innovators, early adopters, early majority, late majority, and laggards [47]. Customers in the same ith market segment are considered as homogeneous and their utility of the jth product configuration is Uij + εij , i = 1, . . . , 5. In such a way, we can compute Uij using (9.11) in our previous study [37]. εij is an error term for each segment–product pair and it follows a normal distribution εij ∼ N(0, 10). The variance of between segments of εij is estimated around four times that of within segments. In order to estimate the hurdle utility hij , a maximum likelihood method is used [20]. Based on the reviews posted on Amazon.com, we can derive the log of past product adoption for all the customers, denoted as D = (V, P, T), in which each tuple i, Pj , t indicates customer vi adopted the product Pj at time t. Let H denotes the set of hurdle utilities. The likelihood of the data maximization problem giventhe model parameters H is expressed as L(D; H) = Pj ∈P log(L(Dj ; H)), subject to the uniindicates the propagation verse of products P. Let Dj δ(i,Pj ) (1 − of Pj . Then L(Dj ; H) = vi ∈V Pr(Pj |i, t) Pr(Pj |i, t))1−δ(i,Pj ) , where t is the time when vi adopted Pj , and δ(i, Pj ) = 1 if vi adopted Pj , and 0 otherwise. Pr(Pj |i, t) is shown in (4). Then the estimation of hij can be solved by the following optimization problem: MaxH s.t.

logL(D; H) hij > 0, vi ∈ V, Pj ∈ P.

(17.1) (17.2)

5) Initialization parameters for the HTGA method, including population size Mp = 1000, crossover probability pc = 0.8, mutation probability pc = 0.2, an orthogonal array L27 (326 ) with only the first 12 columns as shown in Table I, penalty parameter Cp = 10 000, the maximum number of iteration Imax = 1000, the minimum number of iteration Imin = 50, and the minimum number of convergence Cmin = 10. C. Results and Validation The output of the bilevel optimization model includes a set of seed customers S, |S| = n and a set of J + product


Fig. 3.


Convergence for the upper-level model.

configurations that maximize product adoption at the upper level and maximize shared surplus at the lower level. In order to validate the proposed model, both viral product attributes and viral influence attributes are identified. The contributions of viral product attributes are demonstrated by comparing the results with those obtained by methods used in viral marketing. Moreover, by comparing the selected seeds with all the social entities in terms of multiple measures (see Table V), the identified viral influence attributes are proved to be more important and influential than all the social entities as a whole. We set |S| = 20 and 25 to test the model. Figs. 3 and 4 show the convergence of the algorithm in terms of both the lower-level model and the upper-level model. From Fig. 3, we observe that the maximum numbers of adopters for |S| = 20 and 25 are 862 and 1039, respectively. Similarly, from Fig. 4, the maximum numbers of shared surplus for |S| = 20 and 25 are 11 680 and 14 078, respectively. Note that shared surplus is proportional to the number of adopters. Both figures show the convergence in relatively small numbers. On the one hand, the fast convergence shows the efficiency of the HTGA method, and each iteration shows the strategy in (10). On the other hand, it shows the equilibrium between the leader and the follower. The top three optimal chromosomes derived are shown in Table IV. The top two optimal product configurations are the same for two different seed sizes (i.e., |S| = 20 and 25) while the third one is different. To some extent, this shows that some product attributes consistently contribute to product adoption regardless of the influence of social network effects. These product attributes form the set of viral product attributes, including 10 h,