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Pricing of a Three-Stage Supply Chain with a Big Data Company

Abstract

As Big Data is crucial to the competition and growth of companies, this paper considers a multi-stage supply chain with one Big Data company, multiple manufacturers, and one retailer. Demands of different products are dependent on the price and volume of each other, and such information can be obtained by the Big Data company at a cost. Manufacturers will purchase the demand information from the Big Data company and produce and sell products to the retailer. We intend to examine interactions among these firms. Specifically, various power structures are considered, and the equilibrium pricing decisions are analyzed and expressed explicitly through theoretical study. Effects of parameters under different power structures are explored and compared with each other through both special case studies and numerical experiments. This study enables the derivation of managerial insights related to Big Data that are useful for practical applications.

Introduction

With the rapid development of Internet of Things (IoT) and virtual storage, data are being generated, collected, stored, and analyzed on an unprecedented scale. The International Data Corporation has already predicted in 2012 that there would be 40 zettabytes (ZB) of data on the planet by 2020 [1], 44 times greater than that in 2009. Moreover, it is highly anticipated that the amount of data will continue to increase in the coming years at a rate of 50% to 60% annually [2], and the global Big Data market will reach 118.52 billion dollars by 2022 at a compound annual growth rate of 26% during the forecast years from 2014 to 2022 [3]. Therefore, the term of “Big Data” has been coined, and numerous opportunities have been exploited for application and development in the Big Data era.

Particularly, supply chains are flooded with a significant amount of data generated from diverse devices like embedded sensors, cameras, and smartphones. This enables organizations to improve decision making and gain competitive advantages by facilitating supply chain management. Not only individual companies can benefit from the utilization of Big Data, but also other participants in a supply chain. With the help of the information extracted from Big Data, they can obtain insights into customer behaviors and patterns to achieve smarter pricing and high-quality products [4]. By doing so, organizations are likely to better control costs, improve productivity and increase profit margins. As stated by Mcafee and Brynjolfsson [5], productivity rates and profitability of companies can be enhanced by 5% to 6% if Big Data and analytics are incorporated into supply chain management. Therefore, the demand of specialists in information management has been greatly stimulated, and many companies like AG, Oracle, IBM, Microsoft, SAP, EMC, HP and Dell have spent billions of dollars on data analytics and processing [6].

Unfortunately, not all enterprises are able to handle such immense and complicated data by their IT departments [7]. As stated by Hazen et al. [8], supply chain professionals are inundated with data and motivated to use new ways of thinking about how data are produced, organized, and analyzed. Therefore, instead of using traditional and laggard database management apparatus or softwares, numerous enterprises are screening outsourcing their data and using professional Big Data technologies [9]. Under such circumstances, the position of Big Data companies in supply chain management is becoming dramatically important, which is likely to change the structure of original supply chains [10].

Though there exist extensive studies about Big Data in supply chain management, most of them focus on qualitative research such as concepts and characteristics [11,12,13,14]. Quantitative research using mathematical models is relatively laggard. The review in [15] manifests that only 11% of the considered population conduct quantitative research, while Big Data decision models are not viewed strategically at senior levels in most companies based on the results of a recent survey [16]. Olama et al. [17] propose a new Integration Level Model (ILM) tool to measure the readiness of an organization and its data sets against the different levels of data integration. Tan et al. [18] develop an analytic infrastructure based on the deduction graph technique for firms to incorporate their own competence sets with other firms. Hofmann [19] discusses the potential of Big Data on the improvement of supply chain processes and manifests that the Big Data’s “velocity” character had the biggest influences on the bullwhip effect. Liu and Yi [20] consider a supply chain with one retailer and one manufacturer and explore the decision-making issues of Big Data information investment and its effects on supply chain coordination. A low-carbon supply chain with one retailer and one low-carbon manufacturer is analyzed in [21], and four common cost-sharing models and their pricing rules are put forward.

As for quantitative research involving Big Data companies, only a few studies can be found in the existing literature. Liu and Yi [10] consider a supply chain of one manufacturer and one retailer and discuss Big Data information investment decisions and supply chain coordination strategies. The importance of Big Data companies in supply chains is emphasized and the authors believe this factor should be further considered. This study is extended in [22] by exploring information symmetry and asymmetry circumstances. However, Big Data companies are not explicitly involved in the mathematical models of these two studies though the importance is stressed. Liu and Yi [23] further study a three-stage supply chain with one manufacturer, one retailer, and one Big Data company, and a similar problem is studied in [24] with different assumptions. Liu [25] considers an investment decision-making problem of Big Data information for book supply chains with one book publisher, one retailer, and one Data Company. Different from these studies, we extend the application scope by considering multiple manufactures. Such a consideration is based on the fact that many retailers in modern industry have very complex supply systems to distribute nearly 20,000-30,000 individual stock keeping units [26].

In terms of decision making, pricing strategy is a typical and critical decision which has been explored by all the quantitative research involving Big Data companies mentioned above. Besides, more studies without the consideration of Big Data companies have also explored this topic and analyzed the equilibrium under various power structures like Stackelberg or integrated structures. Chiang et al. [27] compare Stackelberg and integrated structures for a system with one manufacturer and one retailer and find that direct marketing helps the manufacturer to improve overall profitability and may not always be detrimental to the retailer under the assumption of homogeneous customer acceptance of online channel. A similar problem is considered in [28] with the assumption that demands depend on channels’ sales efforts. Cattani et al. [29] explore the effect of Internet convenience degree with equal-pricing constraints under Stackelberg and integrated structures. Li et al. [30] explore a Nash bargaining problem when there is only one perishable product with price-dependent stochastic demand. Lu and Chen [31] investigate the interaction between capabilities of introducing Internet channels, pricing strategies and channel structure when the only manufacturer produces the product and acts as the leader. David and Adida [32] study competition and coordination for one manufacture and several symmetric retailers under Stackelberg and integrated structures. Yi et al. [33] point out the optimal pricing and trial-provision strategy in a distribution channel where a software developer and an intermediary agent play a Stackelberg game with the consideration of consideration of a positive network effect. Based on the above review, it can be seen that most of the existing related studies focus on relatively simply structures. Only a few manufacturers are involved, while the literature on multiple manufacturers is far from enough, which limits the applicability of the results.

In this study, pricing decisions are explored for a three-stage supply chain in a Big Data environment. Multiple manufacturers are involved, and the participation of a Big Data company is also considered. The effects of various power structures on equilibrium prices and profits are investigated, and explicit expressions of the equilibrium solutions are provided. In addition, numerical experiments are conducted to examine the effect of relevant parameters on the equilibrium solutions.

The remainder of the paper is organized as follows: Sect. 2 establishes the model considered in this study and introduces the primary notations used. Based on this model, Sect. 3 further discusses in detail the equilibrium solutions of the whole system under different power structures. The impact of parameters on the system is examined in Sect. 4 through numerical studies. Finally, Sect. 5 concludes the paper and outlines future research directions.

Problem Description

In this section, the theoretical model considered in this paper is constructed in Sect. 2.1 first. The corresponding primary notations used are presented in Sect. 2.2 further to facilitate the following discussion.

Model Setup

This study focuses on the pricing strategies of a three-stage supply chain considering Big Data effects. The supply chain consists of one Big Data company, M upstream manufactures and one downstream retailer to fulfill demands of customer orders. Particularly, the Big Data company can obtain demand information (namely, θ) of product m at the cost of \(c_m^b\) and sell it to manufacturer m at the price of \(p_m^b\). Similar to many existing studies [10, 24, 25], the cost and the price for the Big Data company are counted by the number of information which is proportional to customer demands with a ratio of β (called conversion coefficient). In order to produce proper products to meet customers’ demands, manufacturers need to buy demand information from the Big Data company and share this information with the retailer to achieve better sales. Manufacturer m produces product m at the production cost of \(c_m\) and sells to the retailer at a wholesale price of \(\omega _m\). The retailer will buy products from all manufacturers and sells product m at the retail price of \(p_m\).

For each incoming customer order to the retailer, it may contain all of the M products. Demand of product m, i.e., \(q_m\), is related to the prices of all products, and their relationship is characterized by adopting the classic Bertrand competition framework as follows:

$$\begin{aligned} q_m=Q_m+\sum _{m'}\theta _{m'm}p_{m'}, \end{aligned}$$

where Qm represents the potential market size of product m, and matrix θ represents the coefficients between price and demand. Specifically, \(\theta _{m'm}\) reflects the interaction strength between product \(m'\) and product m [34, 35]. \(\theta _{m'm}<0\) implies a supplementary relationship but \(\theta _{m'm}>0\) indicates a substitutable one between the two products. Particularly, we have \(\theta _{mm}<0\), which means a higher price leads to a lower sales quantity for product m.

The profits of the retailer, manufacturer m and the Big Data company are equal to the corresponding revenue minus its cost, namely, \(\Pi _r=\sum _m(p_m-\omega _m)\cdot q_m\), \(\pi _m=(\omega _m-c_m)q_m-p_m^b\beta _mq_m\) and \(\Pi _b=\sum _m(p_m^b-c_m^b)\beta _mq_m\), respectively. Each manufacturer m needs to determine its optimal wholesale price \(\omega ^*_m\) by maximizing its own profit, while the Big Data company and the retailer obtain their corresponding maximum profits by optimizing prices \(p_m^b\) and \(p_m\), respectively. Figure 1 illustrates the model described above.

Fig. 1
figure1

Three-Stage Supply Chain with Big Data: an Illustration

Notations

For convenience, in Table 1, we provide a list of the primary notations used throughout this study. Other notations will be introduced when necessary.

Table 1 Notations throughout this paper

Theoretical Analysis

This section derives the equilibrium prices and profits under various power structures. Through theoretical analysis, explicit expressions of the equilibria are provided for the considered three-stage supply chain.

Big Data Company is the Stackelberg Leader

In this model, the Big Data company is the Stackelberg leader and decides its information prices first. Then each manufacturer m makes a decision on the wholesale price of product m simultaneously based on θ that is known from the Big Data company. Given all of the information, the retailer determines retail prices of the products by maximizing its profit.

Theorem 1

When the Big Data company is the Stackelberg leader, the equilibrium pricing decisions are

$$\begin{aligned} p^{*}= & \; (\theta '+\theta )^{-1}\theta [\theta '(\theta '+\theta )^{-1}\theta +D]^{-1}D[\mathbf {B} c^b+c-(\theta ')^{-1}Q]/2\nonumber \\&+(\theta ')^{-1}[-E+D[\theta '(\theta '+\theta )^{-1}\theta +D]^{-1}\theta (\theta '+\theta )^{-1}]Q, \end{aligned}$$
(1)
$$\begin{aligned} \omega ^{*}= & \; [\theta '(\theta '+\theta )^{-1}\theta +D]^{-1}[D[\mathbf {B} c^b+c\\&-(\theta ')^{-1}Q]/2-\theta (\theta '+\theta )^{-1}Q], \end{aligned}$$
(2)
$$\begin{aligned} p^{b*}= & \; (2\mathbf {B})^{-1}[\mathbf {B}c^b-c-(\theta ')^{-1}Q], \end{aligned}$$
(3)

where D is a diagonal matrix whose principal diagonal elements are the same as \(\theta '(\theta '+\theta )^{-1}\theta\), and B is a diagonal matrix whose principal diagonal elements are β.

Proof

See Appendix 1.

Given the equilibrium pricing decisions provided in Theorem 1, the profit of the retailer is

$$\begin{aligned} \Pi _r= & \; (p-\omega )'\cdot (Q+\theta 'p)\nonumber \\= & \; [(\theta '+\theta )^{-1}(\theta \omega ^*-Q)-\omega ^*]'\cdot [Q+\theta '(\theta '+\theta )^{-1}(\theta \omega ^*-Q)]\nonumber \\= & \; -(Q+\theta '\omega ^*)'(\theta '+\theta )^{-1}\cdot \theta (\theta '+\theta )^{-1}(Q+\theta '\omega ^*)\nonumber \\= & \; -(Q+\theta 'p^*)'(\theta ')^{-1}\cdot (Q+\theta 'p^*). \end{aligned}$$
(4)

The total profits of all manufacturers are

$$\begin{aligned} \sum _m\pi _m= & \; (Q+\theta 'p)'\cdot (\omega -c-\mathbf {B} p^b) \nonumber \\= & \; (Q+\theta '\omega ^*)'(\theta '+\theta )^{-1}\theta '\cdot [\omega ^*-D^{-1}[[\theta '(\theta '+\theta )^{-1}\theta \\ &+D]\omega ^*+\theta (\theta '+\theta )^{-1}Q]]\nonumber \\= & \; -(Q+\theta '\omega ^*)'(\theta '+\theta )^{-1}\theta '\cdot D^{-1}\theta (\theta '+\theta )^{-1}(Q+\theta '\omega ^*)\nonumber \\= & \; -(Q+\theta 'p^*)'\cdot D^{-1}(Q+\theta 'p^*). \end{aligned}$$
(5)

The profit of the Big Data company is

$$\begin{aligned} \Pi _b= & \; (p^b-c^b)'\mathbf {B}\cdot (Q+\theta 'p)\nonumber \\= & \; (Q+\theta 'p^*)'[-D^{-1}-\theta ^{-1}(\theta '+\theta )(\theta ')^{-1}]\cdot (Q+\theta 'p)\nonumber \\= & \; (Q+\theta '\omega ^*)'(\theta '+\theta )^{-1}\theta '[-D^{-1}-\theta ^{-1}\\ &(\theta '+\theta )(\theta ')^{-1}]\cdot \theta (\theta ' +\theta )^{-1}(Q+\theta '\omega ^*)\nonumber \\= & \; -(Q+\theta '\omega ^*)'(\theta '+\theta )^{-1}[E+\theta 'D^{-1}\theta (\theta '+\theta )^{-1}](Q+\theta '\omega ^*)\nonumber \\= & {} -(Q+\theta 'p^*)'[D^{-1}+(\theta ')^{-1}(\theta '+\theta )\theta ^{-1}](Q+\theta 'p^*). \end{aligned}$$
(6)

The total profit of all participants is

$$\begin{aligned} \Pi _{total}= & \; \Pi _r+\sum _m\pi _m+\Pi _b\nonumber \\= & \; -(Q+\theta 'p^*)'[2D^{-1}+2(\theta ')^{-1}+\theta ^{-1}](Q+\theta 'p^*). \end{aligned}$$
(7)

Theorem 1 manifests that the three equilibrium decisions are linearly related to parameters β, ccb and Q. However, each parameter exists in all of the decision variables, which makes the problem highly coupled and complicated. Concerning the individual and the overall profits, it can be observed that their structures are very similar and take the form of \(-(Q+\theta 'p^*)'X(Q+\theta 'p^*)\) where X is an coefficient. This leads to the conjecture that their characters regarding related parameters share a lot of similarities. As can be seen from Eqs. (4)−(7), all of the four profits are multivariate quadratic functions of β, c, cb and Q.

Integrated Supply Chain

In this model, the retailer, the manufacturers and the Big Data company are optimized as a whole to maximize the total profits.

Theorem 2

When the supply chain is operated as a whole, the equilibrium retail prices are

$$\begin{aligned} p^*= & \; (\theta '+\theta )^{-1}[\theta (c+\mathbf {B}c^b)-Q]. \end{aligned}$$

Proof

The profit of the whole supply chain is

$$\begin{aligned} \Pi _{total}= & \; \Pi _r+\sum _m\pi _m+\Pi _b\\= & \; \sum _m(p_m-c_m-\beta _mc^b_m)q_m\\= & \; (p-c-\mathbf {B}c^b)'(Q+\theta 'p).\nonumber \\ \end{aligned}$$

The first- and the second-order derivatives are

$$\begin{aligned} \frac{\partial \Pi _{total}}{\partial p}= & \; Q+(\theta '+\theta )p-\theta (c+\mathbf {B}c^b),\\ \frac{\partial ^2\Pi _{total}}{\partial ^2p}= & \; \theta '+\theta . \end{aligned}$$

Therefore, given the assumption that \(\theta '+\theta\) is negative definite and the results in [36], the second-order derivative matrix is negative semi-definite, and thus the equilibrium solution can be derived by making all equations of the first-order derivatives equal 0.

With the results in Theorem 2, the total profit of all participants is

$$\begin{aligned} \Pi _{total}=&(p-c-\mathbf {B}c^b)'(Q+\theta 'p)\nonumber \\ =&-[(\theta '+\theta )^{-1}\theta '(c+\mathbf {B}c^b)+(\theta '+\theta )^{-1}Q]'[\theta '(\theta '\\ \nonumber&+\theta )^{-1}\theta (c +\mathbf {B}c^b)+\theta (\theta '+\theta )^{-1}Q]. \end{aligned}$$
(8)

Since the values of \(p^b\) and \(\omega\) are independent with the equilibrium, it is possible to properly distribute the profits via \(p^b\) and \(\omega\) so that the participants are more cooperative. Besides, similarly to the analysis of Theorem 1, parameters β, c, cb and Q in Theorem 2 are linearly related to the decision p, but form a quadratic function for the total profit \(\Pi _{total}\).

Special Case

Theorem 1 and 2 in the above subsections provide explicit expressions of equilibrium prices under various power structures. However, the forms of these expressions are complicated due to the considered multi-stage, multi-product system itself. This makes it difficult to make comparisons between different solutions. Given this fact, several important special cases will be discussed first, which can facilitate the solution comparison and insight acquisition. The general cases will be further explored through numerical experiments in the next section.

To begin with, consider the case where only one manufacturer exists. Then the following corollary holds:

Corollary 1

When there is only one manufacturer, that is, M = 1, the equilibrium solution is as shown in Table 2.

Similarly, it can be proved that if the manufacturers have little dependence on each other, the equilibrium solutions can be shown as Corollary 2:

Corollary 2

When the demands of products are independent, that is, \(\theta _{m'm}=0,m'\ne m\), the equilibrium solution is shown in Table 3.

It can be observed that, the retail price under the integrated structure is smaller than that under the BDC-Stackelberg structure, but the total profit is greater. As will be shown in Sect. 4, these characteristics are possessed by much more general problem settings as well. Besides, under the BDC-Stackelberg structure, profit structures of the manufacturers, the retailer, the Big Data company and the while are very similar. This coincides with the observation from Theorem 1 in Sect. 3.1.

Table 2 Results for \(M=1\)
Table 3 Results for \(\theta _{m'm}=0,m'\ne m\)

Remark 1

It should be noted that for the results obtained in this section, negative profit may occur. The underlying reason is that under each power structure, this paper explores the case where all participants will take part in even though they may suffer losses. Consequently, the negative profit just means the least loss or the best choice of the corresponding participant. For example, consider the currently heating practice of community group purchase as an example. In the initial stage, purchasing platforms usually provide very low prices to compete for more customers, and a lot of money needs to be invested in purchase subsidies. Many similar examples can be enumerated especially under the condition of sharing economy.

Numerical Experiments

The numerical experiments in this section are conducted to examine the effects of relevant parameters on system performance under various power structures. Specifically, Sect. 4.1 presents the detailed experiment design and runs simulations to evaluate performances. The corresponding results are further illustrated and analyzed in Sect. 4.2.

Experiment Design

The main purposes of experiments are to explore the following issues:

  • Experiment #1: effects of costs, namely c and cb;

  • Experiment #2: effects of demand sensitivity to price, namely θ;

  • Experiment #3: effects of potential market size, namely Q.

For the above purposes, without loss of generality, the following setting is adopted as a benchmark: the number of products M is 3; the potential market size of product m, Qm, is randomly drawn from the uniform distribution U(0, 2); the production costs of manufactures c, the information costs of the Big Data company cb, the conversion coefficients β and the coefficients between demands and prices θ are randomly drawn from the normal distribution of N(1, 0.3) and take absolute values to guarantee positivity.

The parameter generation method in the benchmark problem will be adopted for the three experiments if not stipulated otherwise. For each experiment, the tested parameters in the benchmark problem are multiplied by a coefficient changing from 0 to 8 in step sizes of 0.2, while the other parameters in the benchmark problem remain the same.

Computational Results

Given the experiment design described in Sect. 4.1, numerical experiments are conducted, and the corresponding results are illustrated in Fig. 2 to Fig. 5. For the convenience of discussion, legends and axes used in these figures will be uniformly explained here first. The legends of “B” and “I” represent power structures, and their meanings are “Big Data company-Stackelberg” and “Integrated” structures, respectively. The X-axes of “Multiplication Coefficient” stand for the coefficients multiplied with the respective parameters, while the Y-axes of “Profit” stand for the profits of corresponding subjects for each tested scenario.

Experiment #1: Effects of Costs, namely c and c b

As can be observed from Figs. 2 and 3, the four respective subfigures share a lot of similarities with each other in terms of changing trend. This is natural as parameters of both c and cb can affect the system by determining the relevant costs. Meanwhile, there also exist some differences between them when magnitude of the changes is concerned.

For the profits of the manufacturers and the Big Data company in the first and the third subfigures, it can be observed that the corresponding lines are linear or convex. This observation can be detected via their profit functions or derivatives. To be specific, for the Big Data company-Stackelberg structure, its profit functions as well as the first-order and the second-order derivatives can be easily expressed with the help of the results in Sect. 3.1. The expressions show that all manufactures’ profit against c and the Big Data company’s profit against cb are linear functions, while the Big Data company’s profit against c and all manufactures’ profit against cb are multivariate convex functions. For the integrated structure, since pb and \(\omega\) are irrelevant and can take any feasible value, they are set to be 0 in this experiment. Under such an assignment, similar conclusions can be reached as well. This assignment also makes their profits under the integrated structure consist only costs. Therefore, it can be observed that with the change of c or cb, the corresponding profits of all manufacturers or the Big Data company will change more significantly than the other one. Besides, for the impact of c or cb, the profit difference between all manufacturers and the Big Data company under the Big Data company-Stackelberg structure is smaller than that under the integrated structure. This is due to the assignment of pb and \(\omega\) which decides the profit distribution among the participants. Accordingly, in practice, the studied integrated structure can achieve balance between efficiency and fairness by properly managing pb and \(\omega\).

Fig. 2
figure2

Profit for Different c

Similarly, for the profit of the retailer in the second subfigures, the lines are linear or convex, which can also be observed via their profit function mathematically. Besides, the integrated structure show a descending trend. This can be attributed to the concrete setting in this numerical experiment. Since \(p^b\) and \(\omega\) are constant and equal to 0, the participants lack cooperation, and thus increasing in costs is only a burden to the retailer and harms its profit.

Fig. 3
figure3

Profit for Different \(c^b\)

For the total profits of all participants in the fourth subfigures, they tend to ascend with the increase in these two parameters. As specified in Sect. 4.1, the parameters are enlarged by multiplying it with a coefficient. This makes the difference between the two sets of costs greater, and thus is possible to achieve more complementary advantages when pricing decisions are made. In addition, the total profit of the integrated system increases more significantly than that of the Big Data company-Stackelberg structure. The attribute of costs can further differentiate the profit structures of the participants, which contributes to the enlargement of double marginalization effect and thus the decrease of overall profit when separate optimization is conducted. This suggests the importance of designing proper contract to reduce internal losses.

Experiment #2: Effects of Demand Sensitivity to Price, namely \(\theta\)

Figure 4 illustrates the results of relevant profits with changes in the scale of \(\theta\). As can be observed, this figure has some differences from those in Experiment #1. Particularly, the curves are more close to each other especially when \(\theta\) becomes larger and may have the opposite tendency to those in Figs. 2 and 3. Both the considered costs in Experiment #1 and \(\theta\) in Experiment #2 can affect profits. However, costs have a more direct influence, while \(\theta\) needs to resort to demands to work. Therefore, the impact of \(\theta\) is weaker and more complicated.

The profits of all manufacturers and the Big Data company in the first and the third subfigures have very similar characteristics. This is because \(\theta\) affects them in the same way, namely via demand. As the scale of \(\theta\) grows up, their curves tend to be stable. When \(\theta\) is near to 0, the manufacturers are almost independent of each other, and thus their competitions are less. With the increase in \(\theta\), the mutual dependence becomes stronger and will remarkably impact the system. In particular, when \(\theta\) is extraordinarily large and the full potential of the market has nearly been reached, the system can only be improved marginally, that is, the performance seems to be stable.

Fig. 4
figure4

Profit for Different \(\theta\)

The profits of the retailer and the total profits in the second and the fourth subfigures share many similarities. In both subfigures, the integrated structure performs better than the Big Data company-Stackelberg structure, which coincides with intuition. Besides, similar to the first and the third subfigures, the curves tend to decrease when \(\theta\) becomes large. Compared to the profits of the manufacturers and the Big Data company, retailer’s profit seems to be larger under the integrated structure. This is because in addition to \(\theta\), retail prices p are also decisive, and thus the retailer has more opportunities to gain more profits by changing p. In contrast, the decisions of the manufacturers and the Big Data company remain unchanged in this experiment study. As the profit of the retailer is dominant to the other participants, the total profit performs in a similar way to the retailer’s profit.

Experiment #3: Effects of Potential Market Size, namely Q

Figure 5 presents the relationships between related profits and potential market size Q. Compared to the results in Figs. 2, 3 and 4, scales of the four subfigures in Fig. 5 seem relatively greater. This is because even if all of the participants make the same decisions, they can earn more as the overall market size is enlarged.

The profits of all manufacturers and the Big Data company in the first and the third subfigures react to potential market size changes in a similar way. Changes in Q are exogenous variables to the manufacturers and the Big Data company, and their decisions cannot affect the final demands directly. Besides, profit differences among the studied structures in each subfigure is relatively small in comparison to the other subfigures. These suggest that manufacturers and the Big Data company are less impacted by changes in potential market size. Given the illustration of the subfigures and the mathematic analysis of the derivatives, it can be seen that the lines are linear under the integrated structure, which mainly attributes to the setting of \(p^b\) and \(\omega\) in this experiment study. Similarly, the lines are concave under the Big Data company-Stackelberg structure, and this suggests that as the potential market size Q increases, it plays a more important role. 

Fig. 5
figure5

Profit for Different Q

The profits of the retailer and the total profits in the second and the fourth subfigures are similar, and their profits tend to be greater than those in the other subfigures. Particularly, profits under the integrated structure is larger, which is expected as this structure can weaken the effect of double marginalization. This suggests the importance for all participants within a supply chain to be coordinated to satisfy customers’ increasing demands. In addition, as shown by the figure and the profit functions, the lines are concave under both the Big Data company-Stackelberg structure and the integrated structure.

Conclusion

To investigate the impacts of Big Data on supply chain performance, this study considers a supply chain management problem in the Big Data era. A three-stage system with one Big Data company, multiple manufacturers and one retailer is explored. In this system, the information of Big Data is crucial to product demands and the profits of all participants. Both the Big Data company-Stackelberg structure and the integrated structure are considered, and the equilibrium pricing decisions are analyzed and expressed explicitly through theoretical study. Effects of parameters under different power structures are explored and compared with each other through both special case studies and numerical experiments. This enables the derivative of managerial insights that are useful for practical applications.

Future research can be conducted in a few directions. One is to consider demand and supply uncertainties such as stochastic arrival and product return. Another direction is to design rational contracts to reduce the effects of double marginalization. In addition, cases of partial information also deserve exploration. Given the complicated three-stage structure in this paper, such directions will be much more challenging to be explored. Besides game theory, other tools like fuzzy programming and contractual arrangement will also be considered.

Data Availability

The authors declare that we do not use any data set in this study.

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Funding

This study was funded by Ministry of Education Program in Humanities and Social Sciences (No. 20YJC630226); the New Teacher Research Start-up Foundation of Shenzhen University [No. 00000270 and 000002110167];  the Natural Science Foundation of Guangdong Province (No. 2018A030313938).

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Correspondence to Zelong Yi.

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This work was supported by Ministry of Education Program in Humanities and Social Sciences (Project name: Research on competition and coordination of retailer’s dual-channel supply chains with multiple products, grant No. 20YJC630226); the New Teacher Research Start-up Foundation of Shenzhen University [grant numbers 00000270 and 000002110167]; the Natural Science Foundation of Guangdong Province (No. 2018A030313938).

Appendix

Appendix

Proof of Theorem 1

The profit of the retailer is

$$\begin{aligned} \Pi _r= & \; \sum _m(p_m-\omega _m)\cdot q_m\nonumber \\= & \; \sum _m(p_m-\omega _m)\cdot (Q_m+\sum _{m'}\theta _{m'm}p_{m'})\nonumber \\= & \; (p-\omega )'\cdot (Q+\theta 'p). \end{aligned}$$
(9)

The first-order derivatives of Eq. (9) are

$$\begin{aligned} \frac{\partial \Pi _r}{\partial p}=Q+(\theta '+\theta )p-\theta \omega . \end{aligned}$$
(10)

Similarly, the second-order derivatives are

$$\begin{aligned} \frac{\partial ^2\Pi _r}{\partial ^2p}=\theta '+\theta . \end{aligned}$$

Assume that \(\theta '+\theta\) and \(\mathbf {B}\theta '(\theta '+\theta )^{-1}\theta [\theta '(\theta '+\theta )^{-1}\theta +D]^{-1}D\mathbf {B}\) are negative definite and that the diagonal elements of \(\theta '(\theta '+\theta )^{-1}\theta\) are negative. It can be derived from [36] that the optimal values of \(p^*\) can be obtained by making Eq. (10) equal 0 since the second-order derivative matrix is negative semi-definite under the adopted assumption. The result is shown as follows:

$$\begin{aligned} p^*=(\theta '+\theta )^{-1}(\theta \omega -Q). \end{aligned}$$
(11)

The profit of manufacturer m is

$$\begin{aligned} \pi _m= & \; (\omega _m-c_m)q_m-p_m^b\beta _mq_m\nonumber \\= & \; (\omega _m-c_m-p_m^b\beta _m)q_m\nonumber \\= & \; (\omega _m-c_m-p_m^b\beta _m)(Q_m+\sum _{m'}\theta _{m'm}p_{m'}). \end{aligned}$$
(12)

Let A be the matrix whose principal diagonal elements are \(\omega _m-c_m-p_m^b\beta _m\) and the other elements are 0. Then Eq. (12) can be expressed in vector space as follows:

$$\begin{aligned} \Pi _m= & \; A(Q+\theta 'p), \end{aligned}$$
(13)

where \(\Pi _m=(\pi _1,\pi _2,\ldots ,\pi _M)'\).

Based on Eq. (11), it can be concluded that

$$\begin{aligned} \frac{\partial p^*}{\partial \omega '}= & \; \left( \begin{array}{cccc} \frac{\partial \rho _1}{\partial \omega _1} &{} \frac{\partial \rho _1}{\partial \omega _2} &{} \ldots &{} \frac{\partial \rho _1}{\partial \omega _M} \\ \frac{\partial \rho _2}{\partial \omega _1} &{} \frac{\partial \rho _2}{\partial \omega _2} &{} \ldots &{} \frac{\partial \rho _2}{\partial \omega _M} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ \frac{\partial \rho _M}{\partial \omega _1} &{} \frac{\partial \rho _M}{\partial \omega _2} &{} \ldots &{} \frac{\partial \rho _M}{\partial \omega _M} \\ \end{array} \right) \nonumber \\= & \; (\theta '+\theta )^{-1}\theta . \end{aligned}$$
(14)

Let B and C be the vector whose elements are the principal diagonal of \(A\theta '\frac{\partial \rho ^*}{\partial \omega }\) and \(2\theta '\frac{\partial \rho ^*}{\partial \omega }\), respectively. Then, the first- and the second-order derivatives of Eq. (13) regarding \(\omega\) are

$$\begin{aligned} \frac{\partial \Pi _m}{\partial \omega '}\overset{\bigtriangleup }{=}&\; \left[\frac{\partial \pi _1}{\partial \omega _1},\frac{\partial \pi _2}{\partial \omega _2},\ldots ,\frac{\partial \pi _M}{\partial \omega _M}\right]'\nonumber \\= & \; Q+\theta 'p+B, \end{aligned}$$
(15)
$$\begin{aligned} \frac{\partial ^2\Pi _m}{\partial ^2\omega '}\overset{\bigtriangleup }{=}&\;\left[\frac{\partial ^2\pi _1}{\partial ^2\omega _1},\frac{\partial ^2\pi _2}{\partial ^2\omega _2},\ldots ,\frac{\partial ^2\pi _M}{\partial ^2\omega _M}\right]'\nonumber \\= & \; C. \end{aligned}$$
(16)

Therefore, given the adopted assumption, the optimal wholesale price \(\omega ^*\) can be obtained by making Eq. (15) equal 0, and there exists

$$\begin{aligned} \omega ^*=[\theta '(\theta '+\theta )^{-1}\theta +D]^{-1}[D(c+\mathbf {B} p^b)-\theta (\theta '+\theta )^{-1}Q], \end{aligned}$$
(17)

where D is a diagonal matrix whose principal diagonal elements are the same as \(\theta '(\theta '+\theta )^{-1}\theta\), and \(\mathbf {B}\) is a diagonal matrix whose principal diagonal elements are \(\beta\).

The profit of the Big Data company is

$$\begin{aligned} \Pi _b= & \; \sum _m(p_m^b-c_m^b)\beta _mq_m\nonumber \\= & \; \sum _m(p_m^b-c_m^b)\beta _m(Q_m+\sum _{m'}\theta _{m'm}p_{m'})\nonumber \\= & \; (p^b-c^b)'\mathbf {B}(Q+\theta 'p). \end{aligned}$$
(18)

Since

$$\begin{aligned} \frac{\partial p}{\partial p^{b'}}\overset{\bigtriangleup }{=}&\;\left( \begin{array}{cccc} \frac{\partial p_1}{\partial p_1^b} &{} \frac{\partial p_1}{\partial p_2^b} &{} \ldots &{} \frac{\partial p_1}{\partial p_M^b} \\ \frac{\partial p_2}{\partial p_1^b} &{} \frac{\partial p_2}{\partial p_2^b} &{} \ldots &{} \frac{\partial p_2}{\partial p_M^b} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ \frac{\partial p_M}{\partial p_1^b} &{} \frac{\partial p_M}{\partial p_2^b} &{} \ldots &{} \frac{\partial p_M}{\partial p_M^b} \\ \end{array} \right) \nonumber \\= & \; \frac{\partial p}{\partial \omega '}\cdot \frac{\partial \omega }{\partial p^{b'}}\nonumber \\= & \; (\theta '+\theta )^{-1}\theta \cdot [\theta '(\theta '+\theta )^{-1}\theta +D]^{-1}D\mathbf {B}, \end{aligned}$$
(19)

there exist

$$\begin{aligned} \frac{\partial \Pi _b}{\partial p^b}= & \; \mathbf {B}(Q+\theta 'p)+[(p^b-c^b)'\mathbf {B}\theta '\frac{\partial p}{\partial p^b}]'\nonumber \\= & \; \mathbf {B}(Q+\theta 'p)+\mathbf {B}D[\theta '(\theta '+\theta )^{-1}\theta \\& +D]^{-1}\theta '(\theta '+\theta )^{-1}\theta \mathbf {B}(p^b-c^b),\\ \frac{\partial ^2\Pi _b}{\partial ^2p^b}= & \; \mathbf {B}\theta '\frac{\partial p}{\partial p^b}+(\frac{\partial p}{\partial p^b})'\theta \mathbf {B}.\nonumber \end{aligned}$$
(20)

Let Eq. (20) equal 0, and there exists

$$\begin{aligned} p^{b*}= & {} -\mathbf {B}^{-1}\theta ^{-1}(\theta '+\theta )(\theta ')^{-1}[\theta '(\theta '+\theta )^{-1}\theta +D]D^{-1}(Q+\theta 'p)+c^b\nonumber \\= & {} -\mathbf {B}^{-1}[D^{-1}+\theta ^{-1}(\theta '+\theta )(\theta ')^{-1}](Q+\theta 'p)+c^b\nonumber \\= & {} -\mathbf {B}^{-1}[D^{-1}+\theta ^{-1}(\theta '+\theta )(\theta ')^{-1}]D(c+\mathbf {B}p^b-\omega )+c^b. \end{aligned}$$
(21)

Based on Eqs. (11), (17) and (21), the explicit expressions of the equilibrium solution can be obtained.

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Zhao, Y., Yi, Z. Pricing of a Three-Stage Supply Chain with a Big Data Company. Oper. Res. Forum 2, 55 (2021). https://doi.org/10.1007/s43069-021-00078-9

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Keywords

  • Big data
  • Pricing decision
  • Supply chain management
  • Multiple manufacturers