^{1}

^{2}

^{2}

^{1}

^{2}

Inventory level has a significant impact on the goodwill of products to customers, which seldom becomes the focus of previous studies. In this paper, joint dynamic pricing, advertising, and production decision-making problem is investigated, where the demand rate depends on sales price and goodwill. The inventory and backlog as well as advertisement are considered as goodwill-building factors. The optimal dynamic pricing, advertising, and production policies are derived by using Pontryagin’s maximum principle. Numerical examples are provided to demonstrate the obtained results, and sensitivity analysis of main system parameters is carried out to obtain some managerial insights. We find that when the initial goodwill is relatively high, the firm’s profit first decreases and then increases with respect to the impact intensity of inventory on goodwill; otherwise, the firm always benefits from a higher impact intensity of inventory on goodwill. Furthermore, the optimal production and advertising policies are complementary caused by the feature of inventory-dependent goodwill.

The effect of inventory level on goodwill is extremely common in real situations. For example, in the supermarket, the item with large quantities displayed in the store generates a good impression that customers think the product is salable; on the contrary, the item which is often backlogged causes a terrible impression to customers. Glamorous display in large numbers with the help of modern light and electronic arrangements attracts the people and brings more customers for purchasing the items [

Considering that inventory level has a motivating effect on customers, previous researches pay more attention to the situation that inventory level has a direct effect on demand. In some retail contexts, stocking large quantities of inventory not only improve service levels but also stimulate demand. For instance, it is usually observed in the supermarket that display of the consumer goods in large quantities attracts more customers and generates higher demand [

The definitions of the mechanism of how the inventory affects demand put forward by researches are quite different. For example, Urban [

Considering the situation of inventory-level-dependent goodwill, firms then take marketing and operation strategies to maximize profit. Pricing, as one of the most important strategies of the modern firms, plays an increasingly important role in sales and marketing. Managers can control prices to influence demand in the short run because of the price-sensitive of customers. The investigation of Marn et al. [

There are a long history and extensive literature on models of combining advertising and pricing since the pioneering work of Dorfman and Steiner [

In the aforementioned literature for optimal pricing and advertising policies, the studies have been confined to one-time pricing and static advertising that the demand has a direct relationship with the frequency of advertisement. As the development of technology innovation such as the increased availability of demand data and the ease of changing prices, it is possible to make dynamic pricing. Dynamic pricing refers to the strategy that firms can adjust the sales price flexibly according to the demand and supply capacity to maximize the profit. Meanwhile, considering the long-time effects of advertising on reputation, firms can contribute to the accumulation of reputation by advertising and achieve its market share and profitability target in the long run. Using the dynamic control strategies, dynamic pricing and dynamic advertising, is particularly helpful to regulate the sales problem, which is widely studied. Particularly, under the case of isoelastic demand, the impact of discounting and specific adoption effects has been studied intensively in an infinite horizon framework, for example, Sethi et al. [

In Table

Contemporary literature review.

Literature | Advertising | Pricing | Production | Inventory | Backlog | Environment |
---|---|---|---|---|---|---|

Gupta and Vrat [ | Static | |||||

Urban [ | Static | |||||

Lu et al. [ | Static | |||||

Zhang et al. [ | Dynamic | |||||

Baye and Morgan [ | Static | |||||

SeyedEsfahania et al. [ | Static | |||||

Yue et al. [ | Static | |||||

Karray and Martin-Herran [ | Static | |||||

He et al. [ | Dynamic | |||||

Chutani and Sethi [ | Dynamic | |||||

Liu et al. [ | Dynamic | |||||

Feng et al. [ | Dynamic | |||||

This paper | Dynamic |

Focus of the problem studied in our setting is different from the available investigations in the following aspects. Firstly, inventory and backlog are considered as goodwill-building factors like advertisement. Secondly, we consider joint pricing, advertising, and production policies problem for the durable product. Finally, continuous time and dynamic environment are presented in this paper. Taking the aforementioned points into account, we establish a production-inventory model of a monopolistic firm which controls production and sales together and seek the optimal pricing, advertising, and production policies to maximize the total profit under the planning horizon, where the demand rate depends on the sales price and goodwill level. The goodwill level is accumulated by the advertising in the most available literature, while we consider that the inventory and backlog are also goodwill-building factors, which is the focus and contribution of this study from the modeling viewpoint. The questions we are concerned about are as follows: What are the optimal production, pricing, and advertising strategies of the firm when the inventory and backlog affect goodwill? What are the interactions between the strategies? How does the impact intensity of inventory on goodwill affect the firm’s profit? The model proposed is a dynamic optimization problem and can be solved by using Pontryagin’s maximum principle. Numerical examples and sensitivity analysis of main system parameters are provided to demonstrate the obtained results. Several interesting results emerge. It is shown that when the initial goodwill is relatively high, the total profit may decrease firstly and then increase with respect to the impact intensity of inventory on goodwill. When the initial goodwill is relatively low, the firm always benefits from a higher impact intensity of inventory on goodwill. What is more, as the inventory plays a role in promoting goodwill, the optimal production and advertising strategies become strategically complementary.

The paper is organized as follows. In the next section, we formulate the mathematical model of the optimization problem. In Section

In this study, we consider a monopolistic firm that produces a kind of durable item and sells it directly to consumers in a finite planning horizon

We consider that the inventory and backlog are goodwill-building factors. Specifically, the positive inventory level has a positive influence on goodwill, where a higher inventory level induces higher goodwill. Conversely, the negative inventory level, that is, backlog, has a negative influence on goodwill, where a higher backlog level decreases goodwill. Both inventory and backlog levels are denoted by

The goodwill, as we all know, has a close relationship with consumer demand, so the demand rate is supposed to depend not only on price but also on the stock of goodwill. Denote the price charged by the firm at time

The inventory system for the firm is considered along a horizon

The cost of advertising effort,

The firm’s objective is to maximize the sum of cash flow along the planning horizon

So, the firm’s optimization problem can be formulated as

The optimization problem (

In this section, the optimization problem (

For the adjoint variables

The optimal control policies

Because

Note that the boundary solutions yield no profit. Here, we only focus on the interior solutions,

Substituting the optimal control strategies into the dynamic evolution equations of states and adjoint variables, we can get the following four-equation system:

By solving the differential equation (

The optimal pricing, advertising, and production strategies can be, respectively, given by

Value of the notations

In this section, we present some numerical examples to illustrate the theoretical results. The focus of this paper is to study the effect of inventory/backlog on goodwill and the influence on other strategies. So we only change parameter

Demand parameters:

Goodwill parameters:

Inventory parameters:

Production parameter:

Planning horizon:

These parameter values are chosen from the previous studies of dynamic production, pricing, and advertising strategies, which allow for a comprehensive illustration. With the given data, we can get the different time paths of

The optimal production strategy with different values of parameters

The optimal advertising strategy with different values of parameters

The optimal dynamic pricing strategy with different values of parameters

The optimal inventory with different values of parameters

Figure

The optimal demand with different values of parameters

Figure

Figure

Figure

Figure

Next, we study how the total profit varies with different values of parameter

The total profit with respect to

In practice, it is common for a firm to manage production and retail together. This paper is concerned with a decision-making problem for a firm to jointly determine the production rate, sales price, and advertising rate in a dynamic setting. Inventory and backlog as well as advertisement are the goodwill accumulating factors. By solving the corresponding optimal control problem on the basis of Pontryagin’s maximum principle, we obtain the joint optimal dynamic pricing, advertising, and production strategies. Numerical examples are provided to illustrate the effectiveness of the main theoretical results and the solution procedure. Then, sensitive analysis of the key system parameters is intensively investigated and some managerial insight is shown. The theoretical contribution and managerial implication can be summarized as follows. Firstly, this paper provides a framework to research the effect on the joint policy when inventory and backlog are goodwill-building factors. Secondly, the analytical solution of the optimal dynamic strategies serves as a powerful reference to support the managers in making the pricing, advertising, and production decision. Thirdly, several interesting results emerge from our research. Specifically, the total profit may decrease firstly and then increase with respect to the impact intensity of inventory on goodwill when the initial goodwill is relatively high. When the inventory plays a role in promoting goodwill, the optimal production and the advertising policies are strategically complementary.

In this paper, we focus on decision-making considering the influence of inventory on goodwill for a single product monopolist firm. For further studies, we can investigate the differential game problem between manufacturers and retailers under a competitive dynamic environment and explore the impact of the effect of goodwill on supply chain members’ decision. In order to better study the actual business situation, both the goodwill and demand dynamics would naturally take uncertainty into account. Furthermore, it is interesting to investigate the costly price adjustment and defect management [

We can rewrite system (

The four eigenvalues of

The eigenvector of A can be specified as

Therefore, we have

Hence, the optimal goodwill and inventory can be given in Proposition

The data used to support the findings of this study are included within the paper.

The authors declare that no conflicts of interest exist with regard to the publication of this paper.

The work was supported by the National Natural Science Foundation of China (Grant no. 71971152), the Key Scientific Research Projects of Colleges and Universities of Henan Province (no. 20B630002), the Foundation for Excellent Youth Teachers of Colleges and Universities of Henan Province under Grant no. 2019GGJS195, and 2018 Research and Development Fund of Anyang Normal University (no. AYNUKP-2018-B25).