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To optimize the firm’s profit during a finite planning horizon, a dynamic programming model is used to make joint pricing and inventory replenishment decision assuming that customers are loss averse and the firm is risk averse. We model the loss averse customer’s demand using the multinomial choice model. In this choice model, we consider the acquisition and transition utilities widely used by a mental accounting theory which also incorporate the reference price and actual price. Then, we show that there is an optimal inventory policy which is base-stock policy depending on the accumulated wealth in each period.

Joint control of inventory and price has long been widely used for many firms such as Amazon, Dell, and J. C. Penny [

For a better operational decision and a successful marketing campaign, a firm’s inventory decision makers should consider customer’s behavior corresponding to the price set by the firm, carefully. Customer’s behavior significantly influences firm’s revenue so that also the firm’s pricing and replenishment decisions are deeply influenced. The firm’s decision makers should construct a good operational and marketing strategy. When you see repeat-purchase markets, consumers have expectation for the price, which is known as reference prices in prospect theory. Customers perceive fluctuating prices as discounts or overcharges relative to the reference prices formed by the previous prices. Moreover, this perception affects demand and thus firm’s profit. For example, while a price discount might have a positive impact on sales on the short-run, the discounted price might result in the installation of a low price in consumers memory, eroding price expectations and willingness to pay and thereby negatively affecting profitability on the long run. It is important for a firm to understand (

So, in this paper, we consider a multiperiod inventory control model in which a risk averse firm faces loss averse customer’s uncertain demand and makes an inventory replenishment and pricing decision by maximizing the firm’s expected utility.

We will go over the literature separately to compare with our research. First, the literature on the customer’s behavior will be reviewed with respect to the loss aversion. Second, the literature on the firm’s behavior will be reviewed with respect to the risk neutral utility. Then, finally, the literature on the firm’s behavior will be reviewed with respect to the risk aversion.

There have been lots of research papers regarding the customers’ irrational behavior since Barbara L. Fredrickson and Daniel Kahneman won the Nobel prize for their works on the prospect theory. Reference [

Second, we will see some traditional research papers on the risk neutral firm. Traditionally, many research literatures consider a model in which the firm is risk neutral and the customer is not loss averse. Actually, the demand from the customer is just affected by the list price set by the firm and is nonincreasing in the price. Reference [

Finally, we will see the literatures on the risk averse firm. The literature on the risk averse inventory control model is quite limited. Reference [

The dynamic control model is utilized in a wide range of industries [

As reviewed above and summarized in Table

Comparison of our research with other existing researches.

[ |
[ |
[ |
[ |
Our research | |
---|---|---|---|---|---|

Risk averse | √ | √ | |||

Loss averse | √ | √ | √ | ||

Replenishment | √ | √ | √ | √ | |

Pricing | √ | √ | √ | √ |

In this paper, the following assumptions are used.

Unsatisfied demand is allowed to be backlogged. So, the inventory level at the beginning of each period can be negative.

Backlogging is widely used assumption in practice. If the demand is unsatisfied, lots of customers are willing to delay receiving what they want.

Replenishment after ordering at the beginning of each period becomes available instantaneously.

In multiperiod inventory control problem, instantaneous replenishment is fairly good assumption if one period is set up widely enough for the replenishment to arrive in that period.

A function

It is an inventory holding cost if

It is incurred at the end of each period and is convex.

The leftover inventory at the end of each period incurs holding cost. Since shortages of inventory may result in the customer’s cancelation of orders or losses in sale which lead to loss of goodwill or profit even for the firm’s business itself, the unsatisfied demand at the end of each period also incurs some shortage cost. If there is not any leftover or shortage of inventory, there is no incurred cost. As the leftover or shortage of inventory increases, the incurred cost in each period should increase.

We consider a model in which there is a single firm selling single product to multiple customers. First, we will see how the loss averse customers behave given the price set by the firm. Then, we will analyze the risk averse firm’s decision process by considering the loss averse customer’s behavior.

All the customers are homogeneous, which implies that customer’s decision is identically and independently distributed. The customer’s demand is basically influenced by the selling price the firm offers to the customer in each period. Also, each customer’s purchasing decision depends on the tradeoff between the selling price and a reference price. As mentioned in [

So far, we see a mathematical expression for the loss averse customer’s decision process. Now, we will see the mathematical expression for the risk averse firm’s decision process.

For the risk averse firm’s decision process, the risk is measured using the increasing and concave utility function and the first derivative of this concave function is decreasing. So, the marginal gain is less than the marginal loss with respective of the same amount of money. Also, as mentioned in [

Extending the consumption model in [

Equation (

Given

In this section, we characterize the firm’s optimal inventory control policy. First, we need to show that

Suppose that the customer is loss averse such that the demand function is (

Suppose that the customer is loss averse such that the demand function is (

We can verify the result of Proposition

In this section, we provide a numerical example with time horizon 4 to show how our model actually works and how the expected utility objectives will change over the various risk averse factors and various loss averse factors.

To consider a firm’s risk aversion, an exponential utility function,

Interestingly, for some numerical instances, the optimal base-stock increases as the firm’s risk aversion increases. For this phenomenon, please see Figure

Let

Optimal base-stock for various firm’s risk aversion

When loss aversion is 1.3

When loss aversion is 1.7

In this paper, we analyze the multiperiod dynamic inventory control problem in which there are a risk averse firm selling single product and many loss averse customers. As mentioned in Introduction, there are lots of research papers considering only loss averse customer or considering only risk averse firm’s strategy. In this paper, we consider dynamic mathematical modeling in which both loss averse customers and risk averse firm are incorporated. Loss averse customers are Multinomial Logit modeled using the acquisition utility and transition utility relative to the reference price. The reference price in each period is considered as a convex combination of the actual price and the reference price at the very previous period. To capture the firm’s risk aversion, we incorporate the firm’s consumption, saving, and borrowing decisions as well as inventory replenishment and pricing decisions. Then, we show that there exists an optimal base-stock inventory policy depending on accumulated wealth in each period.

For the future research, the following can be considered:

One could incorporate various systemic biases in modeling the decisions of the customers such as regret [

It would consider the case where the customers are strategic; for example, they make an intertemporal purchase decision [

In this research, we consider single market in which a firm plays. For the comprehensive view, market competition could be considered so that one can see the effect of heterogenous markets on the firm’s decision and performance.

One who might be interested in behavioral operations, marketing, and promotion strategy together with choice model could use our result as a foundation for one’s future research.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Seungbeom Kim’s work was supported by the Hongik University new faculty research support fund. Jinpyo Lee’s work was supported by 2015 Hongik University Research Fund. Minjae Park’s work was supported by 2015 Hongik University Research Fund. Minjae Park’s research was also supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2014R1A1A2053679). These research funds do not lead to any conflicts of interest regarding the publication of this manuscript.